Notes on Hilbert Spaces Ivan Avramidi New Mexico Tech Socorro, NM 87801

September 2000

Ivan Avramidi: Notes of Hilbert Spaces

2

Notation Logic

A =⇒ B A ⇐= B iff A ⇐⇒ B ∀x ∈ X ∃x ∈ X

A implies B A is implied by B if and only if A implies B and is implied by B for all x in X there exists an x in X such that

Sets and Functions (Mappings) x∈X x 6∈ X {x ∈ X | P (x)} A⊂X X \A A X ×Y f :X→Y f (X) χA ∅ N Z Q R R+ C

x is an element of the set X x is not in X the set of elements x of the set X obeying the property P (x) A is a subset of X complement of A in X closure of set A Cartesian product of X and Y mapping (function) from X to Y range of f characteristic function of the set A empty set set of natural numbers (positive integers) set of integer numbers set of rational numbers set of real numbers set of positive real numbers set of complex numbers

Vector Spaces H ⊕G H∗ Rn Cn l2 lp

direct sum of H and G dual space vector space of n-tuples of real numbers vector space of n-tuples of complex numbers space of square summable sequences space of sequences summable with p-th power

Ivan Avramidi: Notes of Hilbert Spaces

3

Normed Linear Spaces ||x|| xn −→ x w xn −→ x

norm of x (strong) convergence weak convergence

Function Spaces

supp f H ⊗G C0 (Rn ) C(Ω) C k (Ω) C ∞ (Ω) D(Rn ) L1 (Ω) L2 (Ω) Lp (Ω) H m (Ω) C0 (V, Rn ) C k (V, Ω) C ∞ (V, Ω) D(V, Rn ) L1 (V, Ω) L2 (V, Ω) Lp (V, Ω) H m (V, Ω)

support of f tensor product of H and G space of continuous functions with bounded support in Rn space of continuous functions on Ω space of k-times differentiable functions on Ω space of smooth (infinitely diffrentiable) functions on Ω space of test functions (Schwartz class) space of integrable functions on Ω space of square integrable functions on Ω space of functions integrable with p-th power on Ω Sobolev spaces space of continuous vector valued functions with bounded support in Rn space of k-times differentiable vector valued functions on Ω space of smooth vector valued functions on Ω space of vector valued test functions (Schwartz class) space of integrable vector valued functions functions on Ω space of square integrable vector valued functions functions on Ω space of vector valued functions functions integrable with p-th power on Ω Sobolev spaces of vector valued functions

Linear Operators Dα L(H, G) H ∗ = L(H, C)

differential operator space of bounded linear transformations from H to G space of bounded linear functionals (dual space)

4

Ivan Avramidi: Notes of Hilbert Spaces

1

Metric Spaces

Definition 1 A metric space is a set X and a mapping d : X × X → R, called a metric, which satisfies: i) ii)

d(x, y) ≥ 0 d(x, y) = 0 ⇐⇒

x=y

(1.1) (1.2)

iii)

d(x, y) = d(y, x)

(1.3)

iv)

d(x, y) ≤ d(x, z) + d(z, y)

(1.4)

Definition 2 A sequence {xn }∞ n=1 , of elements of a metric space (X, d) is said n→∞ to converge to an element x ∈ X if d(x, xn ) −→ 0. Definition 3 Let (X, d) be a metric space. a) the set B(y, r) = {x ∈ X | d(x, y) < r} is called the open ball of radius r about y; the set B(y, r) = {x ∈ X | d(x, y) ≤ r} is called the closed ball of radius r about y; the set S(y, r) = {x ∈ X | d(x, y) = r} is called the sphere of radius r about y; b) a set O ⊂ X is called open if ∀y ∈ O ∃r > 0: B(y, r) ⊂ O; c) a set N ⊂ X is called a neighborhood of y ∈ N if ∃r > 0: B(y, r) ⊂ N ; d) a point x ∈ X is a limit point of a set E ⊂ X if ∀r > 0: B(x, r) ∩ (E \ {x}) = ∅, i.e. if E contains points other than x arbitrarily close to x; e) a set F ⊂ X is called closed if F contains all its limit points; f ) x ∈ G ⊂ X is called an interior point of G if G is a neighborhood of x. g) The intersection S of all closed sets containing S ⊂ X is called the closure of S. The closure of S ⊂ X is the smallest set containg S. h) The interior of S, S ◦ , is the set of interior points. It is the largest open set contained in S. i) The boundary of S is the set ∂S = S \ S ◦ . Theorem 1 Let (X, d) be a metric space. a) A set O is open iff X \ O is closed d

b) xn −→ x iff ∀ neighborhood N of x ∃m: n ≥ m implies xn ∈ N ; c) the set of interior points of a set is open;

5

Ivan Avramidi: Notes of Hilbert Spaces

d) the union of a set and all its limit points is closed; e) a set is open iff it is a neighborhood of each of its points. f ) The union of any number of open sets is open. g) The intersection of finite number of open sets is open. h) The union of finite number of closed sets is closed. i) The intersection of any number of closed sets is closed. j) The empty set and the whole space are both open nd closed. Theorem 2 A subset S of a metric space X is closed iff every convergent sequence in S has its limit in S, i.e. {xn }∞ n=1 , xn ∈ S, xn → x

=⇒

x∈S

(1.5)

Theorem 3 The closure of a subset S of a metric space X is the set of limits of all convergent sequences in S,i.e. S = {x ∈ X | ∃xn ∈ S : xn → x}.

(1.6)

Definition 4 A subset, Y ⊂ X, of a metric space (X, d) is called dense if d

∀x ∈ X ∃ {yn }∞ n=1 , yn ∈ Y ,: yn −→ x. Theorem 4 Let S be a subset in a metric space X. Then the following conditions are equivalent: a) S is dense in X, b) S = X, c) every non-empty subset of X contains an element of S. Definition 5 A metric space X is called separable if it has a countable dense set. Definition 6 A subset S ⊂ X of a metric space X is called compact if every sequence {xn } in S contains a convergent subsequence whose limit belongs to S. Theorem 5 Compact sets are closed and bounded. Definition 7 A sequence {xn }∞ n=1 , of elements of a metric space (X, d) is called a Cauchy sequence if ∀ > 0 ∃N : n, m ≥ N implies d(xn , xm ) < . Proposition 1 Any convergent sequence is Cauchy. Definition 8 A metric space in which all Cauchy sequences converge is called complete.

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Ivan Avramidi: Notes of Hilbert Spaces

Definition 9 A mapping f : X → Y from a metric space (X, d) to a metric ρ space (Y, ρ) is called continuous at x if f (xn ) −→ f (x) ∀{xn }∞ n=1 , xn ∈ X, d

xn −→ x, i.e. the image of a convergent sequence converges to the image of the limit. Definition 10 A bijection (one-to-one onto mapping) h : X → Y from (X, d) to (Y, ρ) is called isometry if it preserves the metric, i.e. ρ(h(x), h(y)) = d(x, y)

∀x, y ∈ X

(1.7)

Proposition 2 Any isometry is continuous. Theorem 6 If (X, d) is an incomplete metric space, it is possible to find a complete metric space (X, d) so that X is isometric to a dense subset of X.

2

Vector Spaces

Definition 11 A complex vector space is a nonempty set V with two operations: + : V × V → V and · : C × V → V that satisfy the following conditions: ∀x, y, z ∈ V i) ii)

x+y =y+x (x + y) + z = x + (y + z)

(2.8) (2.9)

iii) iv)

∃0 ∈ V : ∀x ∈ V : x + 0 = x ∀x ∈ V ∃(−x) ∈ V : x + (−x) = 0

(2.10) (2.11)

v)

∀α, β ∈ C, ∀x, y ∈ V α(βx) = (αβ)x

(2.12)

vi) vii) viii)

(α + β)x = αx + βx α(x + y) = αx + αy 1·x=x

(2.13) (2.14) (2.15)

A real vector space is defined similarly. Examples (Function Spaces).

Let Ω ⊂ Rn be an open subset of Rn .

1. P (Ω) is the space of all polynomials of n variables as functions on Ω. 2. C(Ω) is the space of all continuous complex valued functions on Ω. 3. C k (Ω) is the space of all complex valued functions with continuous partial derivatives of order k on Ω. 4. C ∞ (Ω) is the space of all infinitely differentiable complex valued (smooth) functions on Ω.

7

Ivan Avramidi: Notes of Hilbert Spaces

Example (Sequence Spaces (lp -Spaces)). Let p ≥ 1. lp is the space of all infinite sequences {zn }∞ n=1 of complex numbers such that ! p1 ∞ X |zn |p < ∞. (2.16) n=1

Definition 12 Let V be a complex vector space and let x1 , . . . xk ∈ V and α1 , . . . , αk ∈ C. A vector x = α1 x1 + · · · αk xk is called a linear combination of x1 , . . . xk . Definition 13 A finite collection of vectors x1 , . . . , xk is called linearly independent if k X

αi xi = 0

⇐⇒

αi = 0 ,

i = 1, 2, . . . , k .

(2.17)

i=1

An arbitrary colection of vectors B = {xn }∞ n=1 is called linearly independent if every finite subcollection is linearly independent. A collection of vectors which is not linearly independent is called linearly dependent. Definition 14 Let B ⊂ V be a subset of a vector space V . Then span B is the set of all finite linear combinations of vectors from B ( k ) X span B = αi xi xi ∈ B, αi ∈ C, k ∈ N . (2.18) i=1

Proposition 3 Let B ⊂ V be a subset of a vector space V . Then span B is a subspace of V . Definition 15 A set of vectors B ⊂ V is called a basis of V (or a base of V ) if B is linearly independent and span B = V . If ∃ a finite basis in V , then V is called finite dimensional vector space. Otherwise V is called infinite dimensional vector space. Proposition 4 The number of vectors in any basis of a finite dimensional vector space is the same. Definition 16 The number of vectors in a basis of a finite dimensional vector space is called the dimension of V , denoted by dim V .

3

Normed Linear Spaces

Definition 17 A normed linear space is a vector space, V , over C (or R) and a mapping || · || : V → R, called a norm, that satisfies: i) ||v|| ≥ 0 ∀v ∈ V ii) ||v|| = 0 ⇐⇒ v=0 iii) ||αv|| = |α| ||v|| ∀v ∈ V, ∀α ∈ C iv)

||v + w|| ≤ ||v|| + ||w||

∀v, w ∈ V

(3.19) (3.20) (3.21) (3.22)

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Ivan Avramidi: Notes of Hilbert Spaces

Examples. 1. Norms in Rn : ||x||2 ||x||1

k X

= =

x2i

! 12

i=1 n X

|xi |

(3.23) (3.24)

i=1

||x||∞

=

max 1≤i≤n {|xi |}

2. A norm in Cn k X

||z|| =

|zi |2

! 12

(3.25)

(3.26)

i=1

3. Let Ω ⊂ Rn be a closed bounded subset of Rn and dx = dx1 · · · dxn be a measure in Rn . Norms in C(Ω) can be defined by ||f ||∞ = sup |f (x)|

(3.27)

x∈Ω

||f ||p =

Z

p1 |f (x)| dx

(3.28)

|f (x)| dx

(3.29)

! p1

(3.30)

p

Ω

||f ||1 =

Z Ω

4. A norm in lp ||z|| =

k X

|zi |p

i=1

5. A norm in l∞ ||z|| = sup |zn |

(3.31)

n∈N

Proposition 5 A normed linear space (V, || · ||) is a metric space (V, d) with the induced metric d(v, w) = ||v − w||. Convergence, open and closed sets, compact sets, dense sets, completeness, in a normed linear space are defined as in a metric space in the induced metric. Definition 18 A normed linear space is complete if it is complete as a metric space in the induced metric. Definition 19 A complete normed linear space is called the Banach space.

9

Ivan Avramidi: Notes of Hilbert Spaces

Definition 20 A bounded linear transformation from a normed linear space (V, || · ||V ) to a normed linear space (W, || · ||W ) is a mapping T : V → W that satisfies: i) T (αv + βw) = αT (v) + βT (w),

∀v, w ∈ V, ∀α, β ∈ C;

ii) ||T (v)||V ≤ C||v||W for some C ≥ 0. iii) the number ||T || =

||T (v)||W ||v||V v∈V,v6=0 sup

(3.32)

is called the norm of T . Theorem 7 Any bounded linear tranformation between two normed linear spaces is continuous. Theorem 8 A bounded linear transformation, T : V → W , from a normed linear space (V, || · ||V ) to a complete normed linear space (W, || · ||W ) can be uniquely extended to a bounded linear transformation, T , from the completion V of V to (W, || · ||W ). The extension of T preserves the norm ||T || = ||T ||.

4

Remarks on Lebesgue Integral

Definition 21 Characteristic function of a set A ⊂ X is a mapping χA : X → {0, 1} defined by 1, if x ∈ A χA (x) = (4.33) 0, if x 6∈ A

Definition 22 For a non-zero function f : Rn → R, the set, supp f , of all points x ∈ Rn for which f (x) 6= 0 is called the support of f , i.e. supp f = {x ∈ Rn |f (x) 6= 0} .

(4.34)

Clearly, supp χA = A. Definition 23 Let I be a semi-open interval in Rn defined by I = {x ∈ Rn | ak ≤ xk < bk , k = 1, . . . , n}

(4.35)

for some ak < bk . The measure of the set I is defined to be µ(I) = (b1 − a1 ) · · · · (bn − an ) .

(4.36)

The Lebesgue integral of a characteristic function of the set I is defined by Z χI dx = µ(I) . (4.37)

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Ivan Avramidi: Notes of Hilbert Spaces

Definition 24 A finite linear combination of characteristic functions of semiopen intervals N X αk χIk (4.38) f= k=1

is called a step function. Definition 25 The Lebesgue integral of a step function is defined by linearity Z X N N X αk χIk dx = αk µ(Ik ) . (4.39) k=1

k=1

Definition 26 A function f : Rn → R is Lebesgue integrable if ∃ a sequence of step functions {fk } such that f'

∞ X

fk ,

(4.40)

k=1

which means that two conditions are satisfied ∞ Z X a) |fk | dx < ∞

(4.41)

k=1

b)

f (x) =

∞ X

fk (x) ∀x ∈ Rn such that

k=1

∞ X

|fk (x)| < ∞ . (4.42)

k=1

The Lebesgue integral of f is then defined by Z ∞ Z X f dx = fk dx

(4.43)

k=1

Proposition 6 The space, L1 (Rn ), of all Lebesgue integrable functions on Rn R is a vector space and is a linear functional on it. R R a) If f, g ∈ L1 (Rn ) and f ≤ g, then f dx ≤ gdx. R R b) If f ∈ L1 (Rn ), then |f | ∈ L1 (Rn ) and | f dx| ≤ |f |dx.

Theorem 9

Theorem 10 If {fk } is a sequence of integrable functions and f'

∞ X

fk ,

(4.44)

k=1

then Z

f=

∞ Z X

k=1

fk ,

(4.45)

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Ivan Avramidi: Notes of Hilbert Spaces

Definition 27 The L1 -norm in L1 (Rn ) is defined by Z ||f || = |f |dx

(4.46)

Definition 28 A function f is called a null function is it is integrable and ||f || = 0. Two functions f and g are said to be equivalent if f − g is a null function. Definition 29 The equivalence class of f ∈ L1 (Rn ), denoted by [f ], is the set of all functions equivalent to f . Remark. Strictly speaking, to make L1 (Rn ) a normed space one has to consider instead of functions the classes of equivalent functions. Definition 30 A set X ⊂ Rn is called a null set (or a set of measure zero) if its characteristic function is a null function. Theorem 11

a) Every countable set is a null set.

b) A countable union of null sets is a null set. c) Every subset of a null set is a null set. Definition 31 Two integrable functions, f, g ∈ L1 (Rn ), are said to be equal almost everywhere, f = g a.e., if the set of all x ∈ Rn for which f (x) 6= g(x) is a null set. Theorem 12 f = g a.e

⇐⇒

||f − g|| =

Z

|f − g| = 0

(4.47)

Theorem 13 The space L1 (Rn ) is complete.

5

Geometry of Hilbert Space

Definition 32 A complex vector space V is called an inner product space (or a pre-Hilbert space if there is a mapping (·, ·) : V × V → C, called an inner product, that satisfies: ∀x, y, z ∈ V, ∀α ∈ C: (x, x) ≥ 0

(5.48)

ii) iii) iv)

(x, x) = 0 ⇐⇒ x=0 (x, y + z) = (x, y) + (x, z) (x, αy) = α(x, y)

(5.49) (5.50) (5.51)

v)

(x, y) = (y, x)∗

(5.52)

i)

Definition 33 Let V be an inner product pace.

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Ivan Avramidi: Notes of Hilbert Spaces

i) Two vectors x, y ∈ V are said to be orthogonal if (x, y) = 0; ii) A collection, {xi }N i=1 , of vectors in V is called an orthonormal set if (xi , xj ) = δij , i.e. (xi , xj ) = 1 if i = j and (xi , xj ) = 0 if i 6= j. Theorem p 14 Every inner product space is a normed linear space p with the norm ||x|| = (x, x) and a metric space with the metric d(x, y) = (x − y, x − y). Theorem 15 (Pythagorean Theorem.) Let V be an inner product space and {xn }N n=1 be an orthonormal set in V . Then ∀x ∈ V ||x||2 =

N X

n=1

2 N X |(x, xn )|2 + x − (x, xn )xn

(5.53)

n=1

Theorem 16 (Bessel inequality.) Let V be an inner product space and {xn }N n=1 be an orthonormal set in V . Then ∀x ∈ V ||x||2 ≥

N X

|(x, xn )|2

(5.54)

n=1

Theorem 17 (Schwarz inequality.) Let V be an inner product space. Then ∀x, y ∈ V |(x, y)| ≤ ||x|| ||y||. (5.55) Theorem 18 (Parallelogram Law) Let V be an inner product space. Then ∀x, y ∈ V ||x + y||2 + ||x − y||2 = 2||x||2 + 2||y||2 . (5.56) Definition 34 A sequence {xn } of vectors in an inner product space V is called strongly convergent to x ∈ V , denoted by xn → x, if n→0

||xn || −→ 0

(5.57) w

and weakly convergent to x ∈ V , denoted by xn → x, if n→0

(xn − x, y) −→ 0

∀y ∈ V .

(5.58)

Theorem 19 a) xn → x b)

w

=⇒

w

xn → x

xn → x and ||xn || → ||x||

(5.59) =⇒

xn → x

(5.60)

Definition 35 A complete inner product space is called a Hilbert space. Definition 36 A linear transformation U : H1 → H2 from a Hilbert space H1 onto the Hilbert space H2 is called unitary if if it preserves the inner product, i.e. ∀x, y ∈ H1 (U x, U y)H2 = (x, y)H1 . (5.61)

13

Ivan Avramidi: Notes of Hilbert Spaces

Definition 37 Two Hilbert spaces H1 and H2 are said to be isomorphic if there is a unitary linear transformation U from H1 onto H2 . Definition 38 Let H1 and H2 be Hilbert spaces. The direct sum H1 ⊕ H2 of Hilbert spaces H1 and H2 is the set of ordered pairs z = (x, y) with x ∈ H1 and y ∈ H2 with inner product (z1 , z2 )H1 ⊕H2 = (x1 , x2 )H1 + (y1 , y2 )H2

6

(5.62)

Examples of Hilbert Spaces

1. Finite Dimensional Vectors. CN is the space of N -tuples x = (x1 , . . . , xN ) of complex numbers. It is a Hilbert space with the inner product (x, y) =

N X

x∗n yn .

(6.63)

n=1

2. Square Summable Sequences of Complex Numbers. l2 is the space of sequences of complex numbers x = {xn }∞ n=1 such that ∞ X

|xn |2 < ∞ .

(6.64)

n=1

It is a Hilbert space with the inner product (x, y) =

∞ X

x∗n yn .

(6.65)

n=1

3. Square Integrable Functions on R. L2 (R) is the space of complex valued functions such that Z |f (x)|2 dx < ∞ . (6.66) R

It is a Hilbert space with the inner product Z (f, g) = f ∗ (x)g(x) dx

(6.67)

R

4. Square Integrable Functions on Rn . Let Ω be an open set in Rn (in particular, Ω can be the whole Rn ). The space L2 (Ω) is the set of complex valued functions such that Z |f (x)|2 dx < ∞ , (6.68) Ω

where x = (x1 , . . . , xn ) ∈ Ω and dx = dx1 · · · dxn . It is a Hilbert space with the inner product Z (f, g) = f ∗ (x)g(x) dx (6.69) Ω

14

Ivan Avramidi: Notes of Hilbert Spaces

5. Square Integrable Vector Valued Functions. Let Ω be an open set in Rn (in particular, Ω can be the whole Rn ) and V be a finite-dimensional vector space. The space L2 (V, Ω) is the set of vector valued functions f = (f1 , . . . , fN ) on Ω such that N Z X |fi (x)|2 dx < ∞ . (6.70) i=1

Ω

It is a Hilbert space with the inner product (f, g) =

N Z X

fi∗ (x)gi (x) dx

(6.71)

Ω

i=1

6. Sobolev Spaces. Let Ω be an open set in Rn (in particular, Ω can be the whole Rn ) and V a finite-dimensional complex vector space. Let C m (V, Ω) be the space of complex vector valued functions that have partial derivatives of all orders less or equal to m. Let α = (α1 , . . . , αn ), α ∈ N, be a multiindex of nonnegative integers, αi ≥ 0, and let |α| = α1 + · · · + αn . Define ∂ |α| f. n · · · ∂xα n

(6.72)

∀α, |α| ≤ m, ∀i = 1, . . . , N, ∀x ∈ Ω .

(6.73)

Dα f =

1 ∂xα 1

Then f ∈ C m (V, Ω) iff |Dα fi (x)| < ∞

The space H m (V, Ω) is the space of complex vector valued functions such that Dα f ∈ L2 (V, Ω) ∀α, |α| ≤ m, i.e. such that N Z X i=1

|Dα fi (x)|2 dx < ∞

∀α, |α| ≤ m .

(6.74)

Ω

It is a Hilbert space with the inner product (f, g) =

X

N Z X

α, |α|≤m i=1

(Dα fi (x))∗ Dα gi (x) dx

(6.75)

Ω

Remark. More precisely, the Sobolev space H m (V, Ω) is the completion of the space defined above.

7

Projection Theorem

Definition 39 Let M be a closed subspace of a Hilbert space H. The set, M ⊥ , of vectors in H which are orthogonal to M is called the othogonal complement of M .

15

Ivan Avramidi: Notes of Hilbert Spaces

Theorem 20 A closed subspace of a Hilbert space and its orthogonal complement are Hilbert spaces. Theorem 21 Let M be a closed subspace of a Hilbert space H. Then ∀x ∈ H ∃ a unique element z ∈ M closest to x. Theorem 22 (Projection Theorem.) Let M be a closed subspace of a Hilbert space H. Then ∀x ∈ H ∃z ∈ M and ∃w ∈ M ⊥ such that x = z + w. That is H = M ⊕ M⊥

(7.76)

Remark. The set, L(H, H 0 ), of linear transformations from a Hilbert space H to H 0 is a Banach space under the norm ||T || =

sup ||T x||H 0

(7.77)

||x||H =1

Definition 40 The space H ∗ = L(H, C) of linear transformations from a Hilbert space H to C is called the dual space of H. The elements of H ∗ are called continuous linear functionals. Theorem 23 (Riesz Lemma) Let H be a Hilbert pace. Then ∀T ∈ H ∗ ∃yT ∈ H such that ∀x ∈ H T (x) = (yT , x),

8

and

||T ||H ∗ = ||yT ||H

(7.78)

Orthonormal Bases

Definition 41 Let S be an orthonormal set in a Hilbert pace H. If there is no other orthonormal set that contains S as a proper subset, then S is called orthonormal basis (or complete orthonormal system) for H. Theorem 24 Every Hilbert space has an othonormal basis. Theorem 25 Let S = {xα }α∈A be an orthonormal basis for a Hilbert space H. Then ∀y ∈ H X y = (xα , y)xα (8.79) α∈A

||y||2

=

X

|(xα , y)|2

(8.80)

α∈A

Definition 42 Let S = {xα }α∈A be an orthonormal basis for a Hilbert space H. The coefficients (xα , y) are called the Fourier coefficients of y ∈ H with respect to the basis S. Definition 43 A metric space which has a countable dense subset is said to be separable. Theorem 26 A Hilbert space H is separable iff it has a countable orthonormal basis S. If S contains finite number, N , of elements, then H is isomorphic to CN . If S contains countably many elements, then H is isomorphic to l2 .

16

Ivan Avramidi: Notes of Hilbert Spaces

9

Tensor Products of Hilbert Spaces

Let H1 and H2 b Hilbert spaces. For each ϕ1 ∈ H1 and ϕ2 ∈ H2 let ϕ1 ⊗ ϕ2 denote the conjugate bilinear form on H1 × H2 defined by (ϕ1 ⊗ ϕ2 )(ψ1 , ψ2 ) = (ψ1 , ϕ1 )H1 (ψ2 , ϕ2 )H2

(9.81)

where ψ1 ∈ H1 and ψ2 ∈ H2 . Let E be the set of finite linear combinations of such bilinear forms. An inner product on E can be defined by (ϕ ⊗ ψ, η ⊗ µ)E = (ϕ, η)H1 (ψ, µ)H2

(9.82)

(with ϕ, η ∈ H1 and ψ, µ ∈ H2 ) and extending by linearity on E. Definition 44 The tensor product H1 ⊗ H2 of the Hilbert paces H1 and H2 is defined to be the completion of E under the inner product defined above. Theorem 27 Let H1 and H2 be Hilbert spaces. If {ϕk } and {ψl } are orthonormal bases for H1 and H2 respectively, then {ϕk ⊗ ψl } is anorthonormal basis for the tensor product H1 ⊗ H2 . Fock Spaces. Let H be a Hilbert space and let H0

=

C

(9.83)

Hn

=

H ⊗ ··· ⊗ H | {z }

(9.84)

n

denote the n-fold tensor product of H. The space n F (H) = ⊕∞ n=0 H

(9.85)

is called the Fock space over H. Fock space F (H) is separable if H is separable. For example, if H = L2 (R), then an element ψ ∈ F (H) is a sequence of functions ψ = {ψo , ψ1 (x), ψ(x1 , x2 ), ψ3 (x1 , x2 , x3 ), . . .} so that ||ψ|| = |ψ0 |2 +

∞ Z X

n=1

|ψn (x1 , . . . , xn )|2 dx1 . . . dxn < ∞

(9.86)

(9.87)

Rn

Let Pn be the permutation group of n elements and let {ϕk } be a basis for H. Each σ ∈ Pn defines a permutation σ(ϕk1 ⊗ · · · ⊗ ϕkn ) = ϕkσ(1) ⊗ · · · ⊗ ϕkσ(n) .

(9.88)

By linearity this can be extended to a bounded operator on H n , so one can define 1 X Sn = σ (9.89) n! σ∈Pn

17

Ivan Avramidi: Notes of Hilbert Spaces

An =

1 X ε(σ)σ n!

(9.90)

σ∈Pn

where

1, ε(σ) = −1

if σ is even

(9.91)

if σ is odd

Finally, the Boson (symmetric) Fock space is defined by n Fs (H) = ⊕∞ n=0 Sn H

(9.92)

and the Fermion (antisymmetric) Fock space is defined by n Fa (H) = ⊕∞ n=0 An H

(9.93)

In the case H = L2 (R), ψn ∈ Sn H n is a function of n variables symmetric under any permutations of variables, and ψn ∈ An H n is a function of n variables that is odd function under interchanges of any two variables.

References [1] M. Reed and B. Simon, Methods of Mathematical Physics, vol. I Functional Analysis, (New York: Academic Press, 1972) [2] R. D. Richtmyer, Principles of Advanced Mathematical Physics, vol. I, (Berlin: Springer, 1985) [3] L. Debnath and P. Mikusi´ nski, Introduction to Hilbert Spaces with Applications, (Boston: Academic Press, 1990)

September 2000

Ivan Avramidi: Notes of Hilbert Spaces

2

Notation Logic

A =⇒ B A ⇐= B iff A ⇐⇒ B ∀x ∈ X ∃x ∈ X

A implies B A is implied by B if and only if A implies B and is implied by B for all x in X there exists an x in X such that

Sets and Functions (Mappings) x∈X x 6∈ X {x ∈ X | P (x)} A⊂X X \A A X ×Y f :X→Y f (X) χA ∅ N Z Q R R+ C

x is an element of the set X x is not in X the set of elements x of the set X obeying the property P (x) A is a subset of X complement of A in X closure of set A Cartesian product of X and Y mapping (function) from X to Y range of f characteristic function of the set A empty set set of natural numbers (positive integers) set of integer numbers set of rational numbers set of real numbers set of positive real numbers set of complex numbers

Vector Spaces H ⊕G H∗ Rn Cn l2 lp

direct sum of H and G dual space vector space of n-tuples of real numbers vector space of n-tuples of complex numbers space of square summable sequences space of sequences summable with p-th power

Ivan Avramidi: Notes of Hilbert Spaces

3

Normed Linear Spaces ||x|| xn −→ x w xn −→ x

norm of x (strong) convergence weak convergence

Function Spaces

supp f H ⊗G C0 (Rn ) C(Ω) C k (Ω) C ∞ (Ω) D(Rn ) L1 (Ω) L2 (Ω) Lp (Ω) H m (Ω) C0 (V, Rn ) C k (V, Ω) C ∞ (V, Ω) D(V, Rn ) L1 (V, Ω) L2 (V, Ω) Lp (V, Ω) H m (V, Ω)

support of f tensor product of H and G space of continuous functions with bounded support in Rn space of continuous functions on Ω space of k-times differentiable functions on Ω space of smooth (infinitely diffrentiable) functions on Ω space of test functions (Schwartz class) space of integrable functions on Ω space of square integrable functions on Ω space of functions integrable with p-th power on Ω Sobolev spaces space of continuous vector valued functions with bounded support in Rn space of k-times differentiable vector valued functions on Ω space of smooth vector valued functions on Ω space of vector valued test functions (Schwartz class) space of integrable vector valued functions functions on Ω space of square integrable vector valued functions functions on Ω space of vector valued functions functions integrable with p-th power on Ω Sobolev spaces of vector valued functions

Linear Operators Dα L(H, G) H ∗ = L(H, C)

differential operator space of bounded linear transformations from H to G space of bounded linear functionals (dual space)

4

Ivan Avramidi: Notes of Hilbert Spaces

1

Metric Spaces

Definition 1 A metric space is a set X and a mapping d : X × X → R, called a metric, which satisfies: i) ii)

d(x, y) ≥ 0 d(x, y) = 0 ⇐⇒

x=y

(1.1) (1.2)

iii)

d(x, y) = d(y, x)

(1.3)

iv)

d(x, y) ≤ d(x, z) + d(z, y)

(1.4)

Definition 2 A sequence {xn }∞ n=1 , of elements of a metric space (X, d) is said n→∞ to converge to an element x ∈ X if d(x, xn ) −→ 0. Definition 3 Let (X, d) be a metric space. a) the set B(y, r) = {x ∈ X | d(x, y) < r} is called the open ball of radius r about y; the set B(y, r) = {x ∈ X | d(x, y) ≤ r} is called the closed ball of radius r about y; the set S(y, r) = {x ∈ X | d(x, y) = r} is called the sphere of radius r about y; b) a set O ⊂ X is called open if ∀y ∈ O ∃r > 0: B(y, r) ⊂ O; c) a set N ⊂ X is called a neighborhood of y ∈ N if ∃r > 0: B(y, r) ⊂ N ; d) a point x ∈ X is a limit point of a set E ⊂ X if ∀r > 0: B(x, r) ∩ (E \ {x}) = ∅, i.e. if E contains points other than x arbitrarily close to x; e) a set F ⊂ X is called closed if F contains all its limit points; f ) x ∈ G ⊂ X is called an interior point of G if G is a neighborhood of x. g) The intersection S of all closed sets containing S ⊂ X is called the closure of S. The closure of S ⊂ X is the smallest set containg S. h) The interior of S, S ◦ , is the set of interior points. It is the largest open set contained in S. i) The boundary of S is the set ∂S = S \ S ◦ . Theorem 1 Let (X, d) be a metric space. a) A set O is open iff X \ O is closed d

b) xn −→ x iff ∀ neighborhood N of x ∃m: n ≥ m implies xn ∈ N ; c) the set of interior points of a set is open;

5

Ivan Avramidi: Notes of Hilbert Spaces

d) the union of a set and all its limit points is closed; e) a set is open iff it is a neighborhood of each of its points. f ) The union of any number of open sets is open. g) The intersection of finite number of open sets is open. h) The union of finite number of closed sets is closed. i) The intersection of any number of closed sets is closed. j) The empty set and the whole space are both open nd closed. Theorem 2 A subset S of a metric space X is closed iff every convergent sequence in S has its limit in S, i.e. {xn }∞ n=1 , xn ∈ S, xn → x

=⇒

x∈S

(1.5)

Theorem 3 The closure of a subset S of a metric space X is the set of limits of all convergent sequences in S,i.e. S = {x ∈ X | ∃xn ∈ S : xn → x}.

(1.6)

Definition 4 A subset, Y ⊂ X, of a metric space (X, d) is called dense if d

∀x ∈ X ∃ {yn }∞ n=1 , yn ∈ Y ,: yn −→ x. Theorem 4 Let S be a subset in a metric space X. Then the following conditions are equivalent: a) S is dense in X, b) S = X, c) every non-empty subset of X contains an element of S. Definition 5 A metric space X is called separable if it has a countable dense set. Definition 6 A subset S ⊂ X of a metric space X is called compact if every sequence {xn } in S contains a convergent subsequence whose limit belongs to S. Theorem 5 Compact sets are closed and bounded. Definition 7 A sequence {xn }∞ n=1 , of elements of a metric space (X, d) is called a Cauchy sequence if ∀ > 0 ∃N : n, m ≥ N implies d(xn , xm ) < . Proposition 1 Any convergent sequence is Cauchy. Definition 8 A metric space in which all Cauchy sequences converge is called complete.

6

Ivan Avramidi: Notes of Hilbert Spaces

Definition 9 A mapping f : X → Y from a metric space (X, d) to a metric ρ space (Y, ρ) is called continuous at x if f (xn ) −→ f (x) ∀{xn }∞ n=1 , xn ∈ X, d

xn −→ x, i.e. the image of a convergent sequence converges to the image of the limit. Definition 10 A bijection (one-to-one onto mapping) h : X → Y from (X, d) to (Y, ρ) is called isometry if it preserves the metric, i.e. ρ(h(x), h(y)) = d(x, y)

∀x, y ∈ X

(1.7)

Proposition 2 Any isometry is continuous. Theorem 6 If (X, d) is an incomplete metric space, it is possible to find a complete metric space (X, d) so that X is isometric to a dense subset of X.

2

Vector Spaces

Definition 11 A complex vector space is a nonempty set V with two operations: + : V × V → V and · : C × V → V that satisfy the following conditions: ∀x, y, z ∈ V i) ii)

x+y =y+x (x + y) + z = x + (y + z)

(2.8) (2.9)

iii) iv)

∃0 ∈ V : ∀x ∈ V : x + 0 = x ∀x ∈ V ∃(−x) ∈ V : x + (−x) = 0

(2.10) (2.11)

v)

∀α, β ∈ C, ∀x, y ∈ V α(βx) = (αβ)x

(2.12)

vi) vii) viii)

(α + β)x = αx + βx α(x + y) = αx + αy 1·x=x

(2.13) (2.14) (2.15)

A real vector space is defined similarly. Examples (Function Spaces).

Let Ω ⊂ Rn be an open subset of Rn .

1. P (Ω) is the space of all polynomials of n variables as functions on Ω. 2. C(Ω) is the space of all continuous complex valued functions on Ω. 3. C k (Ω) is the space of all complex valued functions with continuous partial derivatives of order k on Ω. 4. C ∞ (Ω) is the space of all infinitely differentiable complex valued (smooth) functions on Ω.

7

Ivan Avramidi: Notes of Hilbert Spaces

Example (Sequence Spaces (lp -Spaces)). Let p ≥ 1. lp is the space of all infinite sequences {zn }∞ n=1 of complex numbers such that ! p1 ∞ X |zn |p < ∞. (2.16) n=1

Definition 12 Let V be a complex vector space and let x1 , . . . xk ∈ V and α1 , . . . , αk ∈ C. A vector x = α1 x1 + · · · αk xk is called a linear combination of x1 , . . . xk . Definition 13 A finite collection of vectors x1 , . . . , xk is called linearly independent if k X

αi xi = 0

⇐⇒

αi = 0 ,

i = 1, 2, . . . , k .

(2.17)

i=1

An arbitrary colection of vectors B = {xn }∞ n=1 is called linearly independent if every finite subcollection is linearly independent. A collection of vectors which is not linearly independent is called linearly dependent. Definition 14 Let B ⊂ V be a subset of a vector space V . Then span B is the set of all finite linear combinations of vectors from B ( k ) X span B = αi xi xi ∈ B, αi ∈ C, k ∈ N . (2.18) i=1

Proposition 3 Let B ⊂ V be a subset of a vector space V . Then span B is a subspace of V . Definition 15 A set of vectors B ⊂ V is called a basis of V (or a base of V ) if B is linearly independent and span B = V . If ∃ a finite basis in V , then V is called finite dimensional vector space. Otherwise V is called infinite dimensional vector space. Proposition 4 The number of vectors in any basis of a finite dimensional vector space is the same. Definition 16 The number of vectors in a basis of a finite dimensional vector space is called the dimension of V , denoted by dim V .

3

Normed Linear Spaces

Definition 17 A normed linear space is a vector space, V , over C (or R) and a mapping || · || : V → R, called a norm, that satisfies: i) ||v|| ≥ 0 ∀v ∈ V ii) ||v|| = 0 ⇐⇒ v=0 iii) ||αv|| = |α| ||v|| ∀v ∈ V, ∀α ∈ C iv)

||v + w|| ≤ ||v|| + ||w||

∀v, w ∈ V

(3.19) (3.20) (3.21) (3.22)

8

Ivan Avramidi: Notes of Hilbert Spaces

Examples. 1. Norms in Rn : ||x||2 ||x||1

k X

= =

x2i

! 12

i=1 n X

|xi |

(3.23) (3.24)

i=1

||x||∞

=

max 1≤i≤n {|xi |}

2. A norm in Cn k X

||z|| =

|zi |2

! 12

(3.25)

(3.26)

i=1

3. Let Ω ⊂ Rn be a closed bounded subset of Rn and dx = dx1 · · · dxn be a measure in Rn . Norms in C(Ω) can be defined by ||f ||∞ = sup |f (x)|

(3.27)

x∈Ω

||f ||p =

Z

p1 |f (x)| dx

(3.28)

|f (x)| dx

(3.29)

! p1

(3.30)

p

Ω

||f ||1 =

Z Ω

4. A norm in lp ||z|| =

k X

|zi |p

i=1

5. A norm in l∞ ||z|| = sup |zn |

(3.31)

n∈N

Proposition 5 A normed linear space (V, || · ||) is a metric space (V, d) with the induced metric d(v, w) = ||v − w||. Convergence, open and closed sets, compact sets, dense sets, completeness, in a normed linear space are defined as in a metric space in the induced metric. Definition 18 A normed linear space is complete if it is complete as a metric space in the induced metric. Definition 19 A complete normed linear space is called the Banach space.

9

Ivan Avramidi: Notes of Hilbert Spaces

Definition 20 A bounded linear transformation from a normed linear space (V, || · ||V ) to a normed linear space (W, || · ||W ) is a mapping T : V → W that satisfies: i) T (αv + βw) = αT (v) + βT (w),

∀v, w ∈ V, ∀α, β ∈ C;

ii) ||T (v)||V ≤ C||v||W for some C ≥ 0. iii) the number ||T || =

||T (v)||W ||v||V v∈V,v6=0 sup

(3.32)

is called the norm of T . Theorem 7 Any bounded linear tranformation between two normed linear spaces is continuous. Theorem 8 A bounded linear transformation, T : V → W , from a normed linear space (V, || · ||V ) to a complete normed linear space (W, || · ||W ) can be uniquely extended to a bounded linear transformation, T , from the completion V of V to (W, || · ||W ). The extension of T preserves the norm ||T || = ||T ||.

4

Remarks on Lebesgue Integral

Definition 21 Characteristic function of a set A ⊂ X is a mapping χA : X → {0, 1} defined by 1, if x ∈ A χA (x) = (4.33) 0, if x 6∈ A

Definition 22 For a non-zero function f : Rn → R, the set, supp f , of all points x ∈ Rn for which f (x) 6= 0 is called the support of f , i.e. supp f = {x ∈ Rn |f (x) 6= 0} .

(4.34)

Clearly, supp χA = A. Definition 23 Let I be a semi-open interval in Rn defined by I = {x ∈ Rn | ak ≤ xk < bk , k = 1, . . . , n}

(4.35)

for some ak < bk . The measure of the set I is defined to be µ(I) = (b1 − a1 ) · · · · (bn − an ) .

(4.36)

The Lebesgue integral of a characteristic function of the set I is defined by Z χI dx = µ(I) . (4.37)

10

Ivan Avramidi: Notes of Hilbert Spaces

Definition 24 A finite linear combination of characteristic functions of semiopen intervals N X αk χIk (4.38) f= k=1

is called a step function. Definition 25 The Lebesgue integral of a step function is defined by linearity Z X N N X αk χIk dx = αk µ(Ik ) . (4.39) k=1

k=1

Definition 26 A function f : Rn → R is Lebesgue integrable if ∃ a sequence of step functions {fk } such that f'

∞ X

fk ,

(4.40)

k=1

which means that two conditions are satisfied ∞ Z X a) |fk | dx < ∞

(4.41)

k=1

b)

f (x) =

∞ X

fk (x) ∀x ∈ Rn such that

k=1

∞ X

|fk (x)| < ∞ . (4.42)

k=1

The Lebesgue integral of f is then defined by Z ∞ Z X f dx = fk dx

(4.43)

k=1

Proposition 6 The space, L1 (Rn ), of all Lebesgue integrable functions on Rn R is a vector space and is a linear functional on it. R R a) If f, g ∈ L1 (Rn ) and f ≤ g, then f dx ≤ gdx. R R b) If f ∈ L1 (Rn ), then |f | ∈ L1 (Rn ) and | f dx| ≤ |f |dx.

Theorem 9

Theorem 10 If {fk } is a sequence of integrable functions and f'

∞ X

fk ,

(4.44)

k=1

then Z

f=

∞ Z X

k=1

fk ,

(4.45)

11

Ivan Avramidi: Notes of Hilbert Spaces

Definition 27 The L1 -norm in L1 (Rn ) is defined by Z ||f || = |f |dx

(4.46)

Definition 28 A function f is called a null function is it is integrable and ||f || = 0. Two functions f and g are said to be equivalent if f − g is a null function. Definition 29 The equivalence class of f ∈ L1 (Rn ), denoted by [f ], is the set of all functions equivalent to f . Remark. Strictly speaking, to make L1 (Rn ) a normed space one has to consider instead of functions the classes of equivalent functions. Definition 30 A set X ⊂ Rn is called a null set (or a set of measure zero) if its characteristic function is a null function. Theorem 11

a) Every countable set is a null set.

b) A countable union of null sets is a null set. c) Every subset of a null set is a null set. Definition 31 Two integrable functions, f, g ∈ L1 (Rn ), are said to be equal almost everywhere, f = g a.e., if the set of all x ∈ Rn for which f (x) 6= g(x) is a null set. Theorem 12 f = g a.e

⇐⇒

||f − g|| =

Z

|f − g| = 0

(4.47)

Theorem 13 The space L1 (Rn ) is complete.

5

Geometry of Hilbert Space

Definition 32 A complex vector space V is called an inner product space (or a pre-Hilbert space if there is a mapping (·, ·) : V × V → C, called an inner product, that satisfies: ∀x, y, z ∈ V, ∀α ∈ C: (x, x) ≥ 0

(5.48)

ii) iii) iv)

(x, x) = 0 ⇐⇒ x=0 (x, y + z) = (x, y) + (x, z) (x, αy) = α(x, y)

(5.49) (5.50) (5.51)

v)

(x, y) = (y, x)∗

(5.52)

i)

Definition 33 Let V be an inner product pace.

12

Ivan Avramidi: Notes of Hilbert Spaces

i) Two vectors x, y ∈ V are said to be orthogonal if (x, y) = 0; ii) A collection, {xi }N i=1 , of vectors in V is called an orthonormal set if (xi , xj ) = δij , i.e. (xi , xj ) = 1 if i = j and (xi , xj ) = 0 if i 6= j. Theorem p 14 Every inner product space is a normed linear space p with the norm ||x|| = (x, x) and a metric space with the metric d(x, y) = (x − y, x − y). Theorem 15 (Pythagorean Theorem.) Let V be an inner product space and {xn }N n=1 be an orthonormal set in V . Then ∀x ∈ V ||x||2 =

N X

n=1

2 N X |(x, xn )|2 + x − (x, xn )xn

(5.53)

n=1

Theorem 16 (Bessel inequality.) Let V be an inner product space and {xn }N n=1 be an orthonormal set in V . Then ∀x ∈ V ||x||2 ≥

N X

|(x, xn )|2

(5.54)

n=1

Theorem 17 (Schwarz inequality.) Let V be an inner product space. Then ∀x, y ∈ V |(x, y)| ≤ ||x|| ||y||. (5.55) Theorem 18 (Parallelogram Law) Let V be an inner product space. Then ∀x, y ∈ V ||x + y||2 + ||x − y||2 = 2||x||2 + 2||y||2 . (5.56) Definition 34 A sequence {xn } of vectors in an inner product space V is called strongly convergent to x ∈ V , denoted by xn → x, if n→0

||xn || −→ 0

(5.57) w

and weakly convergent to x ∈ V , denoted by xn → x, if n→0

(xn − x, y) −→ 0

∀y ∈ V .

(5.58)

Theorem 19 a) xn → x b)

w

=⇒

w

xn → x

xn → x and ||xn || → ||x||

(5.59) =⇒

xn → x

(5.60)

Definition 35 A complete inner product space is called a Hilbert space. Definition 36 A linear transformation U : H1 → H2 from a Hilbert space H1 onto the Hilbert space H2 is called unitary if if it preserves the inner product, i.e. ∀x, y ∈ H1 (U x, U y)H2 = (x, y)H1 . (5.61)

13

Ivan Avramidi: Notes of Hilbert Spaces

Definition 37 Two Hilbert spaces H1 and H2 are said to be isomorphic if there is a unitary linear transformation U from H1 onto H2 . Definition 38 Let H1 and H2 be Hilbert spaces. The direct sum H1 ⊕ H2 of Hilbert spaces H1 and H2 is the set of ordered pairs z = (x, y) with x ∈ H1 and y ∈ H2 with inner product (z1 , z2 )H1 ⊕H2 = (x1 , x2 )H1 + (y1 , y2 )H2

6

(5.62)

Examples of Hilbert Spaces

1. Finite Dimensional Vectors. CN is the space of N -tuples x = (x1 , . . . , xN ) of complex numbers. It is a Hilbert space with the inner product (x, y) =

N X

x∗n yn .

(6.63)

n=1

2. Square Summable Sequences of Complex Numbers. l2 is the space of sequences of complex numbers x = {xn }∞ n=1 such that ∞ X

|xn |2 < ∞ .

(6.64)

n=1

It is a Hilbert space with the inner product (x, y) =

∞ X

x∗n yn .

(6.65)

n=1

3. Square Integrable Functions on R. L2 (R) is the space of complex valued functions such that Z |f (x)|2 dx < ∞ . (6.66) R

It is a Hilbert space with the inner product Z (f, g) = f ∗ (x)g(x) dx

(6.67)

R

4. Square Integrable Functions on Rn . Let Ω be an open set in Rn (in particular, Ω can be the whole Rn ). The space L2 (Ω) is the set of complex valued functions such that Z |f (x)|2 dx < ∞ , (6.68) Ω

where x = (x1 , . . . , xn ) ∈ Ω and dx = dx1 · · · dxn . It is a Hilbert space with the inner product Z (f, g) = f ∗ (x)g(x) dx (6.69) Ω

14

Ivan Avramidi: Notes of Hilbert Spaces

5. Square Integrable Vector Valued Functions. Let Ω be an open set in Rn (in particular, Ω can be the whole Rn ) and V be a finite-dimensional vector space. The space L2 (V, Ω) is the set of vector valued functions f = (f1 , . . . , fN ) on Ω such that N Z X |fi (x)|2 dx < ∞ . (6.70) i=1

Ω

It is a Hilbert space with the inner product (f, g) =

N Z X

fi∗ (x)gi (x) dx

(6.71)

Ω

i=1

6. Sobolev Spaces. Let Ω be an open set in Rn (in particular, Ω can be the whole Rn ) and V a finite-dimensional complex vector space. Let C m (V, Ω) be the space of complex vector valued functions that have partial derivatives of all orders less or equal to m. Let α = (α1 , . . . , αn ), α ∈ N, be a multiindex of nonnegative integers, αi ≥ 0, and let |α| = α1 + · · · + αn . Define ∂ |α| f. n · · · ∂xα n

(6.72)

∀α, |α| ≤ m, ∀i = 1, . . . , N, ∀x ∈ Ω .

(6.73)

Dα f =

1 ∂xα 1

Then f ∈ C m (V, Ω) iff |Dα fi (x)| < ∞

The space H m (V, Ω) is the space of complex vector valued functions such that Dα f ∈ L2 (V, Ω) ∀α, |α| ≤ m, i.e. such that N Z X i=1

|Dα fi (x)|2 dx < ∞

∀α, |α| ≤ m .

(6.74)

Ω

It is a Hilbert space with the inner product (f, g) =

X

N Z X

α, |α|≤m i=1

(Dα fi (x))∗ Dα gi (x) dx

(6.75)

Ω

Remark. More precisely, the Sobolev space H m (V, Ω) is the completion of the space defined above.

7

Projection Theorem

Definition 39 Let M be a closed subspace of a Hilbert space H. The set, M ⊥ , of vectors in H which are orthogonal to M is called the othogonal complement of M .

15

Ivan Avramidi: Notes of Hilbert Spaces

Theorem 20 A closed subspace of a Hilbert space and its orthogonal complement are Hilbert spaces. Theorem 21 Let M be a closed subspace of a Hilbert space H. Then ∀x ∈ H ∃ a unique element z ∈ M closest to x. Theorem 22 (Projection Theorem.) Let M be a closed subspace of a Hilbert space H. Then ∀x ∈ H ∃z ∈ M and ∃w ∈ M ⊥ such that x = z + w. That is H = M ⊕ M⊥

(7.76)

Remark. The set, L(H, H 0 ), of linear transformations from a Hilbert space H to H 0 is a Banach space under the norm ||T || =

sup ||T x||H 0

(7.77)

||x||H =1

Definition 40 The space H ∗ = L(H, C) of linear transformations from a Hilbert space H to C is called the dual space of H. The elements of H ∗ are called continuous linear functionals. Theorem 23 (Riesz Lemma) Let H be a Hilbert pace. Then ∀T ∈ H ∗ ∃yT ∈ H such that ∀x ∈ H T (x) = (yT , x),

8

and

||T ||H ∗ = ||yT ||H

(7.78)

Orthonormal Bases

Definition 41 Let S be an orthonormal set in a Hilbert pace H. If there is no other orthonormal set that contains S as a proper subset, then S is called orthonormal basis (or complete orthonormal system) for H. Theorem 24 Every Hilbert space has an othonormal basis. Theorem 25 Let S = {xα }α∈A be an orthonormal basis for a Hilbert space H. Then ∀y ∈ H X y = (xα , y)xα (8.79) α∈A

||y||2

=

X

|(xα , y)|2

(8.80)

α∈A

Definition 42 Let S = {xα }α∈A be an orthonormal basis for a Hilbert space H. The coefficients (xα , y) are called the Fourier coefficients of y ∈ H with respect to the basis S. Definition 43 A metric space which has a countable dense subset is said to be separable. Theorem 26 A Hilbert space H is separable iff it has a countable orthonormal basis S. If S contains finite number, N , of elements, then H is isomorphic to CN . If S contains countably many elements, then H is isomorphic to l2 .

16

Ivan Avramidi: Notes of Hilbert Spaces

9

Tensor Products of Hilbert Spaces

Let H1 and H2 b Hilbert spaces. For each ϕ1 ∈ H1 and ϕ2 ∈ H2 let ϕ1 ⊗ ϕ2 denote the conjugate bilinear form on H1 × H2 defined by (ϕ1 ⊗ ϕ2 )(ψ1 , ψ2 ) = (ψ1 , ϕ1 )H1 (ψ2 , ϕ2 )H2

(9.81)

where ψ1 ∈ H1 and ψ2 ∈ H2 . Let E be the set of finite linear combinations of such bilinear forms. An inner product on E can be defined by (ϕ ⊗ ψ, η ⊗ µ)E = (ϕ, η)H1 (ψ, µ)H2

(9.82)

(with ϕ, η ∈ H1 and ψ, µ ∈ H2 ) and extending by linearity on E. Definition 44 The tensor product H1 ⊗ H2 of the Hilbert paces H1 and H2 is defined to be the completion of E under the inner product defined above. Theorem 27 Let H1 and H2 be Hilbert spaces. If {ϕk } and {ψl } are orthonormal bases for H1 and H2 respectively, then {ϕk ⊗ ψl } is anorthonormal basis for the tensor product H1 ⊗ H2 . Fock Spaces. Let H be a Hilbert space and let H0

=

C

(9.83)

Hn

=

H ⊗ ··· ⊗ H | {z }

(9.84)

n

denote the n-fold tensor product of H. The space n F (H) = ⊕∞ n=0 H

(9.85)

is called the Fock space over H. Fock space F (H) is separable if H is separable. For example, if H = L2 (R), then an element ψ ∈ F (H) is a sequence of functions ψ = {ψo , ψ1 (x), ψ(x1 , x2 ), ψ3 (x1 , x2 , x3 ), . . .} so that ||ψ|| = |ψ0 |2 +

∞ Z X

n=1

|ψn (x1 , . . . , xn )|2 dx1 . . . dxn < ∞

(9.86)

(9.87)

Rn

Let Pn be the permutation group of n elements and let {ϕk } be a basis for H. Each σ ∈ Pn defines a permutation σ(ϕk1 ⊗ · · · ⊗ ϕkn ) = ϕkσ(1) ⊗ · · · ⊗ ϕkσ(n) .

(9.88)

By linearity this can be extended to a bounded operator on H n , so one can define 1 X Sn = σ (9.89) n! σ∈Pn

17

Ivan Avramidi: Notes of Hilbert Spaces

An =

1 X ε(σ)σ n!

(9.90)

σ∈Pn

where

1, ε(σ) = −1

if σ is even

(9.91)

if σ is odd

Finally, the Boson (symmetric) Fock space is defined by n Fs (H) = ⊕∞ n=0 Sn H

(9.92)

and the Fermion (antisymmetric) Fock space is defined by n Fa (H) = ⊕∞ n=0 An H

(9.93)

In the case H = L2 (R), ψn ∈ Sn H n is a function of n variables symmetric under any permutations of variables, and ψn ∈ An H n is a function of n variables that is odd function under interchanges of any two variables.

References [1] M. Reed and B. Simon, Methods of Mathematical Physics, vol. I Functional Analysis, (New York: Academic Press, 1972) [2] R. D. Richtmyer, Principles of Advanced Mathematical Physics, vol. I, (Berlin: Springer, 1985) [3] L. Debnath and P. Mikusi´ nski, Introduction to Hilbert Spaces with Applications, (Boston: Academic Press, 1990)