Summary. In [20] the series of real numbers were investigated. The introduction to Banach and Hilbert Spaces ( [12,13,14]), enables us to arrive at the concept of ...
FORMALIZED
MATHEMATICS
Vol.2,No.5, November–December 1991 Universit´ e Catholique de Louvain
Series in Banach and Hilbert Spaces Jan Popiolek Warsaw University Bialystok
El˙zbieta Kraszewska Warsaw University Bialystok
Summary. In [20] the series of real numbers were investigated. The introduction to Banach and Hilbert Spaces ( [12,13,14]), enables us to arrive at the concept of series in Hilbert Space. We start with the notions: partial sums of series, sum and n-th sum of series, convergent series (summable series), absolutely convergent series. We prove some basic theorems: the necessary condition for a series to converge, Weierstrass’ test, d’Alembert’s test, Cauchy’s test. MML Identifier: BHSP 4.
The notation and terminology used here have been introduced in the following articles: [5], [23], [28], [3], [4], [1], [10], [8], [9], [7], [20], [2], [29], [21], [22], [17], [27], [26], [24], [16], [12], [13], [15], [6], [11], [14], [25], [18], and [19]. For simplicity we adopt the following convention: X denotes a real unitary space, a, b, r denote real numbers, s1 , s2 , s3 denote sequences of X, R1 , R2 , R3 denote sequences of real numbers, and k, n, m denote natural numbers. The scheme Rec Func Ex RUS deals with a real unitary space A, a point B of A, and a binary functor F yielding a point of A and states that: there exists a function f from into the vectors of the vectors of A such that f (0) = B and for every element n of and for every point x of A such that x = f (n) holds f (n + 1) = F(n, x) for all values of the parameters. P Let us consider X, s1 . The functor ( κα=0 s1 (α))κ∈ yields a sequence of X and is defined as follows: P Pκ (0) = s (0) and for every n holds ( (Def.1) ( κα=0 s (α)) 1 1 κ∈ α=0 s1 (α))κ∈ (n+ P 1) = ( κα=0 s1 (α))κ∈ (n) + s1 (n + 1).
Next we state several propositions: P P P (1) ( κα=0 s2 (α))κ∈ + ( κα=0 s3 (α))κ∈ = ( κα=0 (s2 + s3 )(α))κ∈ . Pκ P Pκ (2) ( α=0 s2 (α))κ∈ − ( α=0 s3 (α))κ∈ = ( κα=0 (s2 − s3 )(α))κ∈ . P P (3) ( κα=0 (a · s1 )(α))κ∈ = a · ( κα=0 s1 (α))κ∈ .
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1991 Fondation Philippe le Hodey ISSN 0777–4028
elz˙ bieta kraszewska and jan popiolek
696 (4) (5) Let (Def.2)
( κα=0 (−s1 )(α))κ∈ = −( κα=0 s1 (α))κ∈ . P P P a·( κα=0 s2 (α))κ∈ +b·( κα=0 s3 (α))κ∈ = ( κα=0 (a·s2 +b·s3 )(α))κ∈ . us consider X, s1 . We say that s1 is summable if and only if: P ( κα=0 s1 (α))κ∈ is convergent. P
P
Let us consider X, s1 . Let us assume that s1 is summable. The functor yielding a point of X is defined by: P P (Def.3) s1 = lim(( κα=0 s1 (α))κ∈ ).
P
s1
Next we state several propositions: (6) PIf s2 is summable and s is summable, then s2 + s3 is summable and P P 3 (s2 + s3 ) = s2 + s3 . (7) PIf s2 is summable and s is summable, then s2 − s3 is summable and P P 3 (s2 − s3 ) = s2 − s3 . P P (8) If s1 is summable, then a · s1 is summable and (a · s1 ) = a · s1 . (9) If s1 is summable, then s1 is convergent and lim s1 = 0the vectors of X . (10) If X is a Hilbert space, then s 1 is summable if and only if for every r such that r > 0P there exists k such that for all n, m such that n ≥ k and P m ≥ k holds k( κα=0 s1 (α))κ∈ (n) − ( κα=0 s1 (α))κ∈ (m)k < r. P (11) If s1 is summable, then ( κα=0 s1 (α))κ∈ is bounded. P (12) For all s1 , s2Psuch that for every n holds s2 (n) = s1 (0) holds ( κα=0 (s1 ↑ 1)(α))κ∈ = ( κα=0 s1 (α))κ∈ ↑ 1 − s2 . (13) If s1 is summable, then for every k holds s 1 ↑ k is summable. (14) If there exists k such that s 1 ↑ k is summable, then s1 is summable. P Let us consider X, s1 , n. The functor nκ=0 s1 (κ) yielding a point of X is defined by: Pκ Pn (Def.4) κ=0 s1 (κ) = ( α=0 s1 (α))κ∈ (n).
We now state several propositions: Pκ Pn (15) κ=0 s1 (κ) = ( α=0 s1 (α))κ∈ (n). P0 s1 (κ) = s1 (0). (16) P0 Pκ=0 1 (17) κ=0 s1 (κ) + s1 (1). κ=0 s1 (κ) = P1 (18) κ=0 s1 (κ) = s1 (0) + s1 (1). Pn+1 Pn (19) κ=0 s1 (κ) + s1 (n + 1). κ=0 s1 (κ) = Pn+1 P (20) s1 (n + 1) = κ=0 s1 (κ) − nκ=0 s1 (κ). P P (21) s1 (1) = 1κ=0 s1 (κ) − 0κ=0 s1 (κ). P Let us consider X, s1 , n, m. The functor m κ=n+1 s1 (κ) yielding a point of X is defined by: Pm Pn Pm (Def.5) κ=0 s1 (κ). κ=0 s1 (κ) − κ=n+1 s1 (κ) =
The following propositions are true: Pm Pn Pm (22) κ=0 s1 (κ). κ=0 s1 (κ) − κ=n+1 s1 (κ) = (23)
P0
κ=1+1 s1 (κ)
= s1 (1).
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series in banach and hilbert spaces Pn
= s1 (n + 1). If X is a Hilbert space, then s 1 is summable if and only if for every r such that r > P 0 there exists P k such that for all n, m such that n ≥ k and m ≥ k holds k nκ=0 s1 (κ) − m κ=0 s1 (κ)k < r. (26) If X is a Hilbert space, then s 1 is summable if and only if for every r such that r > P 0 there exists k such that for all n, m such that n ≥ k and m ≥ k holds k m κ=n+1 s1 (κ)k < r. P Let us consider R1 , n. The functor nκ=0 R1 (κ) yields a real number and is defined by: Pκ Pn (Def.6) κ=0 R1 (κ) = ( α=0 R1 (α))κ∈ (n). (24) (25)
κ=n+1+1 s1 (κ)
Let us consider R1 , n, m. The functor m κ=n+1 R1 (κ) yielding a real number is defined by: Pm Pn Pm (Def.7) κ=0 R1 (κ). κ=0 R1 (κ) − κ=n+1 R1 (κ) = P
Let us consider X, s1 . We say that s1 is absolutely summable if and only if: (Def.8) ks1 k is summable. The following propositions are true: (27) If s2 is absolutely summable and s3 is absolutely summable, then s2 +s3 is absolutely summable. (28) If s1 is absolutely summable, then a · s 1 is absolutely summable. (29) If for every n holds ks1 k(n) ≤ R1 (n) and R1 is summable, then s1 is absolutely summable. 1 (n+1)k If for every n holds s1 (n) 6= 0the vectors of X and R1 (n) = ksks and 1 (n)k R1 is convergent and lim R1 < 1, then s1 is absolutely summable. (31) If r > 0 and there exists m such that for every n such that n ≥ m holds ks1 (n)k ≥ r, then s1 is not convergent or lim s1 6= 0the vectors of X . (32) If for every n holds s1 (n) 6= 0the vectors of X and there exists m such that for every n such that n ≥ m holds ksks1 (n+1)k ≥ 1, then s1 is not summable. 1 (n)k
(30)
(33)
(34) (35) (36) (37) (38) (39) (40)
If for every n holds s1 (n) 6= 0the vectors of X and for every n holds and R1 is convergent and lim R1 > 1, then s1 is not R1 (n) = ksks1 (n+1)k 1 (n)k summable. p If for every n holds R1 (n) = n ks1 (n)k and R1 is convergent and lim R1 < 1, then s1 is absolutely summable. p If for every n holds R1 (n) = n ks1 k(n) and there exists m such that for every n such that n ≥ m holds R1 (n) ≥ 1, then s1 is not summable. p If for every n holds R1 (n) = n ks1 k(n) and R1 is convergent and lim R1 > 1, then s1 is not summable. P ( κα=0 ks1 k(α))κ∈ is non-decreasing. P For every n holds ( κα=0 ks1 k(α))κ∈ (n) ≥ 0. P P For every n holds k( κα=0 s1 (α))κ∈ (n)k ≤ ( κα=0 ks1 k(α))κ∈ (n). P P For every n holds k nκ=0 s1 (κ)k ≤ nκ=0 ks1 k(κ).
elz˙ bieta kraszewska and jan popiolek
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κ κ For all n, m holds k( P α=0 s1 (α))κ∈ (m) − ( α=0 s1 (α))κ∈ (n)k ≤ κ |( α=0 ks1 k(α))κ∈ (m) − ( α=0 ks1 k(α))κ∈ (n)|. (42) P For all n, m holds Pn Pm Pn k m κ=0 ks1 k(κ)|. κ=0 ks1 k(κ) − κ=0 s1 (κ)k ≤ | κ=0 s1 (κ) − Pn Pn (43) For all n, m holds k κ=m+1 s1 (κ)k ≤ | κ=m+1 ks1 k(κ)|. (44) If X is a Hilbert space, then if s 1 is absolutely summable, then s1 is summable. Let us consider X, s1 , R1 . The functor R1 · s1 yielding a sequence of X is defined as follows: (Def.9) for every n holds (R1 · s1 )(n) = R1 (n) · s1 (n).
(41)
Pκ
P
P
One can prove the following propositions: (45) R1 · (s2 + s3 ) = R1 · s2 + R1 · s3 . (46) (R2 + R3 ) · s1 = R2 · s1 + R3 · s1 . (47) (R2 R3 ) · s1 = R2 · (R3 · s1 ). (48) (a R1 ) · s1 = a · (R1 · s1 ). (49) R1 · −s1 = (−R1 ) · s1 . (50) If R1 is convergent and s1 is convergent, then R1 · s1 is convergent. (51) If R1 is bounded and s1 is bounded, then R1 · s1 is bounded. (52) If R1 is convergent and s1 is convergent, then R1 · s1 is convergent and lim(R1 · s1 ) = lim R1 · lim s1 . Let us consider R1 . We say that R1 is a Cauchy sequence if and only if: (Def.10) for every r such that r > 0 there exists k such that for all n, m such that n ≥ k and m ≥ k holds |R1 (n) − R1 (m)| < r. One can prove the following propositions: (53) If X is a Hilbert space, then if s 1 is a Cauchy sequence and R1 is a Cauchy sequence, then R1 · s1 is a Cauchy sequence. P P · ( κα=0 s1 (α))κ∈ )(α))κ∈ (n) = (54) PFor every n holds ( κα=0 ((R1 − R1 ↑ 1) P ( κα=0 (R1 · s1 )(α))κ∈ (n + 1) − (R1 · ( κα=0 s1 (α))κ∈ )(n + 1). (55) PFor every n holds P P ·s1 )(α))κ∈ (n+1) = (R1 ·( κα=0 s1 (α))κ∈ )(n+1)−( κα=0 ((R1 ↑ ( κα=0 (R1P 1 − R1 ) · ( κα=0 s1 (α))κ∈ )(α))κ∈ (n). Pn+1 Pκ (56) PFor every n holds P κ=0 (R1 · s1 )(κ) = (R1 · ( α=0 s1 (α))κ∈ )(n + 1) − κ n κ=0 ((R1 ↑ 1 − R1 ) · ( α=0 s1 (α))κ∈ )(κ).
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Received April 1, 1992
