Generalized Fock spaces and the Stirling numbers

arXiv:1804.05254v1 [math.FA] 14 Apr 2018

GENERALIZED FOCK SPACES AND THE STIRLING NUMBERS DANIEL ALPAY AND MOTKE PORAT

Abstract. The Bargmann-Fock-Segal space plays an important role in mathematical physics, and has been extended into a number of directions. In the present paper we imbed this space into a Gelfand triple. The spaces forming the Fr´echet part (i.e. the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and G. Salomon.

AMS Classification. Primary 30H20; Secondary 46E22 Key words: Fock space, topological algebra, Stirling numbers. 1. Introduction The reproducing kernel Hilbert space F1 of entire functions with reproducing kernel ezw is associated with the names of Bargmann, Segal and Fock, and will be called in this paper the Fock space (more precisely, it is the symmetric Fock space associated with C). It plays an important role in stochastic processes, mathematical physics and quantum mechanics. F1 is isometrically included in the Lebesgue space of the 2 plane with weight dA(z) := π1 e−|z| dxdy, and a key feature of F1 is that the adjoint of the operator of multiplication by the complex variable is the operator of differentiation. It is of interest to look at various generalizations of F1 . One approach consists in slightly modifying the weight function, see e.g. the works [15, 20, 22] and another line is to change the kernel (that is the norms of the monomials) in an appropriate way, for instance replacing the exponential by the Mittag-Leffler function in the case of the grey noise theory, see e.g. [21]. Then too the weight is changed, but not always in an explicit way. Here we consider the family (Fm )∞ m=1 of reproducing kernel Hilbert spaces with reproducing Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research. 1

2

D. ALPAY AND M. PORAT

kernel (1.1)

∞ X znωn , km (z, ω) = (n!)m n=0

m = 1, 2, . . .

The space Fm can then be described as the space of all Taylor Peasily ∞ series of the form f (z) = n=0 fn z n for which ∞ X n=0

|fn |2 (n!)m < ∞.

For m = 1, the space is equal to the classical Fock space, and the case m = 2 was defined and studied in [7]. The outline of the paper is as follows. In Section 2 we review some facts on the Mellin transform. In Section 3, using the Mellin transform, we give a geometric characterization of the spaces Fm for m ∈ N. A characterization of these spaces in terms of the adjoint of the operator of multiplication by z and using the Stirling numbers of the second kind is given in Section 4. A related Bargmann transform is defined in Section 5. In Section 6 we define a Gelfand triple T in which we imbed the Fock space. We observe that the intersection ∞ m=1 Fm is a nuclear space and its dual is an algebra of the type introduced in [8]. 2. Preliminaries Let (a, b) a open interval of the real line, and let f and g such that both f (x)xc−1 and g(x)xc−1 are summable on [0, ∞) for c ∈ (a, b). The Mellin transform of f , denoted by M(f ), is given by Z ∞ M(f )(c) := xc−1 f (x)dx, c ∈ (a, b). 0

In particular, the Mellin transform of the function f1 (x) = e−x is the Gamma function: Z ∞ M(f1 )(c) = xc−1 e−x dx = Γ(c), c > 0. 0

The Mellin convolution of f and g is defined by Z ∞ Z ∞ x dt x dt (f ∗ g)(x) := f ( )g(t) = f (t)g( ) , t t t t 0 0

x > 0.

An important relation between the Mellin transform and the Mellin convolution, see e.g. [14, Theorem 3], is given by M(f ∗ g)(c) = (M(f )(c))(M(g)(c)),

c ∈ (a, b).

GENERALIZED FOCK SPACES AND THE STIRLING NUMBERS

3

3. Geometric description of Fm Recall that the Fock space F1 consists of those entire functions f for which ZZ 2 |f (z)|2 e−|z| dA(z) < ∞, C

and is the reproducing kernel Hilbert space with reproducing kernel ezw . In this section we give for m = 2, . . . a geometric characterization for the space ( ) ∞ ∞ X X Fm = f (z) = an z n is entire with |an |2 (n!)m < ∞ n=0

n=0

which is the reproducing kernel Hilbert space with reproducing kernel (1.1), when equipped with the inner product hf, giFm :=

∞ X

fn gn (n!)m , where f (z) =

n=0

∞ X

fn z n , g(z) =

n=0

∞ X

gn z n ,

n=0

for every f, g ∈ Fm . First, we use the properties of the Mellin transform to build the kernels Km (z), which are generalizations of the modified Bessel function of the second order, also called the Macdonald function. Let K1 (x) = e−x and for every integer m > 1 define the function (3.1)

Km (x) := (K1 ∗ · · · ∗ K1 )(x),

x ∈ R+

that is the function K1 (x) Mellin-convoluted m many times with itself. Lemma 3.1. Let m be an integer. The following properties hold: (1) For m > 1, the kernel Km has the integral representations (3.2)

Km (x) =

Z

∞

0

∞

···

Z

Z

(3.3)

Km (x) =

R

···

e−

e

Pm−1 i=1

x xi − Qm−1 i=1

Qm−1

0

i=1

and Z

−

√

m

xi

xi

dx1 · · · dxm−1

P P − m−1 ti ti i=1 ) x( m−1 i=1 e +e

R

dt1 · · · dtm−1 .

(2) The function Km is monotone decreasing in (0, ∞). (3) The Mellin transform of Km is given by M(Km )(x) = Γ(x)m ,

x > 0,

and so (3.4)

Z

0

∞

xn Km (x)dx = (n!)m ,

n ∈ N.

4


Proof. Part 1 is proved by induction on m: if m = 2, we get Z ∞ −x1 − xx Z ∞ 1 e −x/t −t dt = dx1 . K2 (x) = e e t x1 0 0 Suppose formula (3.2) holds for m. Then Z ∞ x dxm −t Km+1 (x) = (Km ∗ e )(x) = Km e−xm xm xm 0 = =

Z Z

∞ 0 ∞ 0

···

Z

···

Z

∞

−

e

0 ∞

−

e

Pm−1 i=1

i=1 xi Q x i=1 xi − m

Pm

i=1

xi

i=1

Qm−1

Qm

0

x

xm xi − Qm−1

x i=1 i

xi

e−xm dx1 · · · dxm xm

dx1 · · · dxm ,

i.e., (3.2) holds for m + 1 and hence for every m > 1. Next, we use (3.2) and the change of variables si = ln(xi ), 1 ≤ i ≤ m − 1, to obtain Z Z P x si − m−1 i=1 e − Pm−1 s i e i=1 ds1 · · · dsm−1 Km (x) = ··· e R

R

√ and by another change of variables ti = si − ln( m x), 1 ≤ i ≤ m − 1, we get Z Z Pm−1 √ P eti +e− i=1 ti ) − m x( m−1 i=1 dt1 · · · dtm−1 . Km (x) = ··· e R

R

From the representation (3.2) it is easily seen that Km (x) is a monotone decreasing function. Finally, the Mellin transform of Km is given by M(Km )(c) = M(f1 )(c) · · · M(f1 )(c) = (Γ(c))m ,

therefore

Z

∞

xc−1 Km (x)dx = (Γ(c))m ,

c > 0,

c > 0.

0

For c = n + 1, we have Z ∞ xn Km (x)dx = (Γ(n + 1))m = (n!)m . 0

In the special case m = 2, we get that Z √ K2 (x) = e− x2 cosh(t) dt, R

x ∈ R+

is the Bessel function of the second kind, see [7]. We now show how the generalized Fock spaces Fm are obtained from the kernels Km (x) in a natural way.


5

Theorem 3.2. For any integer m ≥ 1, the space Fm is equal to the space of all entire functions f : C → C satisfying the condition ZZ (3.5) |f (z)|2 Km (|z|2 )dA(z) < ∞. C

Moreover, the inner product of Fm is given by ZZ ∞ X 1 f (z)g(z)Km (|z|2 )dA(z) = fn gn (n!)m , π C n=0 and Fm has the orthonormal basis

n

zn (n!)m/2

o∞

n=0

f, g ∈ Fm ,

.

Proof. A straightforward computation shows that ZZ Z ∞ Z 2π n k 2 z z Km (|z| )dA(z) = r n einθ r k e−ikθ Km (r 2 )rdθdr C 0 0 Z 2π Z ∞ i(n−k)θ = e dθ r n+k+1Km (r 2 )dr 0 0 Z ∞ = 2πδn,k r 2n+1 Km (r 2 )dr Z0 ∞ du = 2πδn,k un Km (u) 2 0 m = π(n!) δn,k . P∞ P n n Let f = ∞ n=0 gn z be entire functions. Then n=0 fn z and g = ZZ ZZ ∞ X 2 fn gk π f (z)g(z)Km (|z| )dA(z) = z n z k Km (|z|2 )dA(z) C

=

n,k=0 ∞ X

n,k=0

C

m

fn gk δn,k (n!) =

∞ X

fn gn (n!)m ,

n=0

which implies that f ∈ Fm if and only if condition (3.5) holds, i.e., ZZ ∞ 1X 2 m |fn | (n!) = |f (z)|2 Km (|z|2 )dA(z) < ∞ π n=0 C as wanted. Furthermore, the inner product in Fm is then given by ZZ ∞ X 1 m f (z)g(z)Km (|z|2 )dA(z). fn gn (n!) = hf, giFm = π C n=0

6

D. ALPAY AND M. PORAT ǫ n!

< 1 for every n ≥ 0 and hence ! ! ∞ ∞ ∞ ∞ ∞ m n n X X X X X ǫ z ω znωn = ǫm km (z, ω) = ǫm m (n!) n! n=0 m=1 m=1 m=1 n=0 ∞ ∞ X X 1 ǫ zn ωn n n = ω = ǫ · z n! 1 − n!ǫ n! − ǫ n=0 n=0

Remark 3.3. Let 0 < ǫ < 1. Then

and

∞ ∞ X X ǫm ǫ km (z, ω) = e n! − 1 z n ω n . m! m=1 n=0

4. Operator theoretic description of Fm Denote by a the operator of multiplication by z by b differentiation by ∂ . Both a and b are defined on polynomiz, i.e., a = Mz and b = ∂z als and more generally on entire functions. They satisfy the familiar commutation relation [b, a] = ba − ab = I. In the Fock space F1 , a and b are unbounded operators, and satisfy a∗ = b

and

b∗ = a.

This relation is very important, as the Fock space is the only space of entire functions for which a and b are adjoint to each other, see [9]. We generalize this result by presenting a relation between the operators a and b in the space Fm . That gives us another characterization of the space Fm . We first introduce the Stirling numbers of the second kind S(k, n), which appear naturally in the theory of ordering bosons. Definition 4.1 (Stirling numbers of the second kind). For k ∈ N0 and n ∈ N0 , the numbers S(k, n) are defined by the recurrence formula S(k, n) = nS(k − 1, n) + S(k − 1, n − 1),

k, n ≥ 1

with the initial values S(k, 0) = δk,0 and S(k, n) = 0 if k < n. It is well known, see [11, 12], that k

(ab) =

k X n=1

S(k, n)an bn ,

k≥1

and this operator is called the Mellin derivative operator of order k (with c = 0), see [14, Lemma 9].


7

Theorem 4.2. Let m ≥ 1 be an integer. The operators a and (ba)m−1 b are closed densely defined operators on the space Fm and their domains coincide Dom(a) = Dom((ba)m−1 b) = D, where (4.1)

D=

(

f (z) =

∞ X

fn z n :

n=0

∞ X n=0

|fn |2 (n!)m nm < ∞

)

⊆ Fm .

Moreover, the adjoint operator of a in Fm is given by a∗ = (ba)m−1 b,

with Dom(a∗ ) = Dom (ba)m−1 b = D.

Furthermore, let H be a Hilbert space of entire functions in which the polynomials are dense, and let m ∈ N. If the adjoint operator of a in H is equal to the operator (ba)m−1 b, i.e., if "m−1 # n X ∂ ∂ (Mz )∗ = S(m − 1, n)z n n , ∂z n=1 ∂z

then H = Fm and there exists c > 0 for which hf, giH = c · hf, giFm ,

∀f, g ∈ H.

Proof. It is easy to see that P a and (ba)m−1 b are closed, densely defined n operators on Fm . If f (z) = ∞ n=0 fn z ∈ Fm , then f ∈ Dom(a) ⇐⇒ af =

and f ∈ Dom((ba)

m−1

∞ X n=0

fn z n+1 ∈ Fm ⇐⇒

b) ⇐⇒ (ba) ⇐⇒

∞ X n=1

m−1

bf =

∞ X n=1

2 2m

|fn | n

∞ X n=0

|fn |2 ((n + 1)!)m < ∞

fn nm z n−1 ∈ Fm m

((n − 1)!) =

∞ X n=1

|fn |2 (n!)m nm < ∞.

Therefore, Dom(a) = Dom((ba)m−1 b) = D as in (4.1). Next, if g(z) =

∞ X n=0

there exists

h(z) =

gn z n ∈ Dom(a∗ ),

∞ X n=0

hn z n ∈ Fm

8


such that haf, giFm = hf, hiFm for every f ∈ Dom(a). In particular, for f (z) = z n (n ≥ 0), we get gn+1 ((n + 1)!)m = hz n+1 , giFm = hz n , hiFm = hn (n!)m

and hence hn = gn+1(n + 1)m for every n ≥ 0. Thus, h ∈ Fm =⇒ =⇒

∞ X

n=0 ∞ X n=1

∗

2

m

|hn | (n!) =

∞ X n=0

|gn+1 |2 (n + 1)2m (n!)m < ∞

|gn |2 (n!)m nm < ∞ =⇒ g ∈ D,

hence Dom(a ) ⊆ D. Finally, if g ∈ D = Dom((ba)m−1 b), then + *∞ ∞ ∞ X X X m−1 n m n fn (n + 1)m gn+1 (n!)m = hf, (ba) bgi = (n + 1) gn+1 z fn z , n=0

=

∞ X

n=0

n=0

fn gn+1 ((n + 1)!)m =

n=0

*∞ X

fn z n+1 ,

n=0

∞ X

gn z n

n=0

+

= haf, gi,

for every f ∈ D = Dom(a), which proves that g ∈ Dom(a∗ ). Therefore, D ⊆ Dom(a∗ ) and hence Dom(a∗ ) = D. By the previous calculation we also know that a∗ = (ba)m−1 b. Now suppose that H is a Hilbert space which contains all polynomials, and such that a∗ = (ba)m−1 b in H. Then for every f ∈ Dom(a) ∩ H and g ∈ Dom((ba)m−1 b) ∩ H,

(4.2)

haf, giH = hf, (ba)m−1 bgiH

and as both Dom(a) and Dom((ba)m−1 b) contain all polynomials, we apply (4.2) for the choice f (z) = z l , g(z) = z k (k, l ≥ 0), thus hz l+1 , z k iH = haf, giH = hf, (ba)m−1 bgiH

= hz l , k m z k−1 iH = k m hz l , z k−1 iH ,

k, l ≥ 0.

We now prove by induction that for every k ≥ 0 and l ≥ k, hz l+1 , z k iH = 0 :

• If k = 0, we know that hz l+1 , 1iH = 0 for every l ≥ 0. • Assume that for some k ≥ 0 we have hz l+1 , z k iH = 0 for every l ≥ k. Therefore, hz l+2 , z k+1 iH = (k + 1)m hz l+1 , z k iH = 0 for every l ≥ k, which means that hz l+1 , z k+1 iH = 0

for every l ≥ k + 1, as wanted.


9

Thus the family {z k }∞ k=0 is orthogonal in H and one can easily see that hz k , z k iH = k m hz k−1 , z k−1 iH ,

∀k ≥ 1,

which implies that hz k , z k iH = (k!)m h1, 1iH. P∞ P k k To conclude, if f (z) = ∞ k=0 gk z ∈ H, then k=0 fk z and g(z) = hf, giH =

∞ X

k,l=0

k

l

fk gl hz , z iH =

∞ X k=0

fk gk (k!)m h1, 1iH,

i.e., the inner product in H is equal to the one in Fm , up to a positive multiplicative constant c = h1, 1iH. As H is a Hilbert space which contains all the polynomials, it follows that ( ) ∞ ∞ X X H= f = fn z n : hf, f iH = c |fn |2 (n!)m < ∞ = Fm . n=0

n=0

In the previous theorem we proved that Fm is the only Hilbert space which contains all polynomials and in which the adjoint operator of a = Mz is equal to the operator "m−1 # n X ∂ ∂ S(m − 1, n)z n n . (ba)m−1 b = ∂z n=1 ∂z One can see that we have the relations bn a = abn + nbn−1

and ban = an b + nan−1

for every n ∈ N, and in particular the operators a and a∗ do not satisfy the commutation relation. However we have the following result. Proposition 4.3. The commutator of a and a∗ = (ba)m−1 b is equal to (4.3)

[a∗ , a] = I +

m−1 X

(n + 1)S(m, n + 1)an bn .

n=1

Proof. As ∗

a = (ba)

m−1

b=b

m−1 X n=1

S(m − 1, n)an bn ,

10


we have ∗

[a , a] = b

m−1 X n=1

=b

m−1 X n=1

n n

S(m − 1, n)a b a − ab n

=

n=1

=

n

m−1 X

n−1

n=1

S(m − 1, n)an bn ) − ab

S(m − 1, n)an bn + b

S(m − 1, n)an bn +

m−1 X n=1

n=1

S(m − 1, n)a (ab + nb

= (ba − ab) m−1 X

m−1 X

m−1 X n=1

m−1 X n=1

m−1 X n=1

S(m − 1, n)an bn

nS(m − 1, n)an bn−1

nS(m − 1, n)(an b + nan−1 )bn−1

(n + 1)S(m − 1, n)an bn +

m−1 X n=1

n2 S(m − 1, n)an−1bn−1

and as S(m − 1, 1) = S(m − 1, m − 1) = S(m, m) = 1, we have ∗

[a , a] = I + ma

m−1 m−1

b

+

m−2 X n=1

= I + mam−1 bm−1 +

(n + 1)[S(m − 1, n) + (n + 1)S(m − 1, n + 1)]an bn

m−2 X

(n + 1)S(m, n + 1)an bn

n=1

=I+

m−1 X

(n + 1)S(m, n + 1)an bn .

n=1

Finally, a straightforward calculation shows that # "∞ m−1 X m X 2 m k 2 ∗ 2 2 |fn | (n!) n , kaf kFm = ka f kFm + kf kFm + k n=0 k=1

for every f ∈ D, which guarantees that all the terms in the identity are finite. It is tempting to write the last identity (with some abuse of notation) as m−1 X m 2 ∗ 2 2 hf, (ab)k f iFm , kaf kFm = ka f kFm + kf kFm + k k=1 however f ∈ D does not necessarily imply that f ∈ Dom((ab)k ). Remark 4.4. Unlike the situation in the the Fock space, where the adjoint of b is equal to a, in the space Fm we get that the adjoint of b


11

is the operator ∗

b

∞ X

fk z

k=0

k

!

=

∞ X k=0

fk z k+1 . m−1 (k + 1)

5. Generalized Bargmann Transform Recall that the normalized Hermite functions are defined by 2 (n) t2 1 √ e 2 e−t , n ∈ N0 . ηn (t) = π 1/4 2n/2 n! The family {ηn }∞ n=0 is an orthonormal basis of the Lebesgue space L2 (R, dt). Furthermore, see [19, p. 436], the ηn are uniformly bounded by some constant, i.e., ∃C > 0 such that |ηn (t)| ≤ C, for every n ∈ N and t ∈ R. Similarly to the Fock space F1 , there is a a fourth characterization of the space Fm , given by a mapping from L2 (R, dt) into Fm , presented in the following proposition. Proposition 5.1. Let m ≥ 2. For every t ∈ R and z ∈ C define the function (5.1)

hm (z, t) :=

∞ X n=0

zn ηn (t). (n!)m/2

Then, 1. for every t ∈ R, the function hm (·, t) is entire. 2. f ∈ Fm if and only if there exists g ∈ L2 (R, dt), such that Z f (z) = hm (z, t)g(t)dt = hg, hm(z, ·)iL2 (R,dt) . (5.2) R

Proof. Since the functions ηn (t) are all bounded by C, the sum in (5.1) converges and so hm (·, t) is entire. Next, let f (z) = hg, hm (z, ·)iL2 (R,dt) for some g ∈ L2 (R, dt). Then, ! Z X Z ∞ ∞ X zn zn f (z) = η (t)g(t) dt = ηn (t)g(t)dt. m/2 n m/2 (n!) (n!) R R n=0 n=0 As the system {ηn }∞ n=0 forms an orthonormal basis of L2 (R, dt), we have Parseval’s equality 2 Z ∞ Z X ηn (t)g(t)dt = |g(t)|2dt n=0

R

R

12


and hence f ∈ Fm , since 2 Z ∞ X 1 ηn (t)g(t)dt (n!)m = kgk2L2 (R,dt) < ∞. (n!)m/2 R n=0 P∞ n Finally, let f ∈ Fm . It can be written as f (z) = n=0 an z with P∞ 2 m n=0 |an | (n!) < ∞. Setting g(t) =

∞ X

(n!)m/2 an ηn (t),

n=0

we observe that kgk2L2 (R,dt) and finally that hhm (z, ·), giL2(R,dt) =

=

∞ X n=0

∞ X n=0

|an |2 (n!)m < ∞

zn (n!)m/2 an = f (z). (n!)m/2

This characterization of Fm motivates us to consider an associated Bargmann transform. For any g ∈ L2 (R, dt) we define the Bargmann transform of g to be Z ∞ X zn ηn (t)g(t)dt = hg, hm(z, ·)iL2 (R,dt) . Bm (g) := (n!)m/2 R n=0 The mapping Bm : L2 (R, dt) → Fm is unitary; it satisfies Bm (ηn )(z) =

zn (n!)m/2

and kgkL2 (R,dt) = kBm (g)kFm

for every g ∈ L2 (R, dt). Remark 5.2. In case where m = 1, B1 is the well known Bargmann transform and the function h1 (z, t) can be written in closed form as 2 −z 2 /2

h1 (z, t) = e2tz−t

.

When m > 1, finding an explicit closed formula for the function hm (z, t) might involve new generalizations of the exponential function. 6. A Gelfand Triple associated to the family (Fm )m∈Z The reproducing kernel Hilbert spaces {Fm }∞ m=1 , starting from the Fock space F1 , form a decreasing sequence, i.e., F1 ⊃ F2 ⊃ ... ⊃ Fm ⊃ Fm+1 ⊃ ....


13

so it makes sense, in the spirit of the theory of Gelfand triples (as developed for instance in the books [17, 18]) to consider the intersection space F= =

∞ \

m=1

(

Fm

f=

∞ X n=0

an z n such that kf km =

∞ X n=0

|an |2 (n!)m < ∞, ∀m ∈ N

)

which consists of entire functions, and its dual. We consider the dual space of each Fm , with respect to the Fock space F1 . Lemma 6.1. For every m ≥ 1, the dual space of Fm , with respect to F1 is ) ( ∞ X |bn |2 (n!)2−m < ∞ . F2−m := (Fm )′ = b = (bn )n∈N0 : kbk22−m := n=0

Therefore, we have the Gelfand triple

(6.1)

∞ \

m=1

Fm ⊂ F1 ⊂

∞ [

m=1

F2−m .

The inclusion map from Fm into Fm+1 is nuclear, and it follows that ∩∞ echet nuclear space, and in particular a perfect space m=1 F2−m is a Fr´ in the terminology of Gelfand and Shilov; see [18]. The dual space ∪∞ m=1 F2−m has two different set of properties, topological and algebraic; the first follow from the theory of perfect spaces, and the structure algebra comes from the form of the weights. The fact that the product is jointly continuous comes from the theory of reflexive Fr´echet spaces; see [13, IV.26, Theorem 2]. We begin with the topological properties. Although not metrizable, the space ∪∞ m=1 F2−m behaves well with respect to sequences and compactness: (1) A sequence converges in the strong (or weak) topology of the dual if and only if its elements are in one of the spaces F2−m and converges in the topology of the latter; see [18, p. 56]. (2) A subset of ∪∞ m=1 F2−m is compact in the strong topology of the dual if and only if it is included in one the spaces F2−m and compact in the topology of the latter; see [18, p. 58]. These properties allow us to reduce to the Hilbert space setting and sequences the study of continuous functions from a compact metric space

14


into ∪∞ m=1 F2−m . The algebra structure is given by the convolution product (or Cauchy product) defined as follows: (6.2)

a ∗ b :=

n X k=0

ak bn−k

!

,

n∈N0

where a = (an )n∈N0 and b = (bn )n∈N0 belong to the dual. Proposition 6.2. The space ( ) ∞ ∞ 2 [ X |b | n F ′ := F2−m = b = (bn )n∈N0 : ∃m ≥ 1, kbk2−m :=