Int. Journal of Math. Analysis, Vol. 3, 2009, no. 36, 1761 - 1768
Some Vector Sequence Spaces-II K. Chandrasekhara Rao, S. Tamilselvan* and K. Vairamanickam* Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam, India *Mathematics Section, Faculty of Engineering and Technology Annamalai University, Annamalainagar-608002, India vk
[email protected] Abstract In this paper, we present some theorems involved in determining sets and Schauder basis. Matrix transformation of some vector sequence spaces are also studied.
Mathematics Subject Classification: 46A45 Keywords: Banach algebra, Determining sets, Schauder, basis, Matrix Transformation
1
Introduction
Let X denote a commutative Banach algebra with the identity e. Then ae = ea = a ∀ a ∈ X and e = 1. Let Φ be the set of all finite sequences in X. Let xk ∈ X and we write x = {xk } and it is a vector sequence. If S is a vector sequence space then D = {x ∈ S : |x| ≤ 1} ∩ Φ. If E ⊂ Φ then the absolutely convex hull of E is denoted by A. The set E is called a determining set for S if A = D. If S and V are sequence spaces, the set of all infinite matrices A with entries in X is denoted by (S : V ). This paper is devoted to characterization of (l : l∞ ) and (bv0 : l∞ ) by using determining sets. = {x : xk < ∞}. ∞ = {x : supxk < ∞}. bv0 = {x : xk − xk+1 < ∞} c0 = {x : limk→∞ xk < ε, ∀ε > 0} . Also the topological duals ∗ and C0∗ are also find out.
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K. Chandrasekhara Rao, S. Tamilselvan and K. Vairamanickam
Results
Theorem 1: E = (δ 1 , δ 2 , . . . ) is determining set for . Proof: δ n = (0, . . . , 0, 1, 0, . . . ) 1 in the nth place and zeros elsewhere. A = absolutely convex hull of E where E = (δ 1 , δ 2 , . . . ) D = ( the closed unit ball in ) ∩ Φ. Let x ∈ A. ⇒ x = t1 δ 1 + t2 δ 2 + . . . + tm δ m With t1 + t2 + . . . + tm ≤ 1 and ti ∈ X. ⇒ x = t1 (e, 0, 0, . . . ) + t2 (0, e, . . . ) + . . . tm (0, . . . e, 0, . . . ) ⇒ x = (t1 , t2 , . . . tm , 0, 0, . . . ) ⇒ x ∈ Φ. Also,
(1)
|||x||| = |||t1δ 1 + t2 δ 2 + . . . + tm δ m ||| ≤ |||t1δ 1 ||| + |||t2 δ 2 ||| + . . . + |||tm δ m ||| = |||t1||| |||δ 1||| + |||t2 ||| |||δ 2||| + . . . + |||tm ||| |||δ m ||| Note that |||δ 1 ||| = (1, 0, . . . ) = 1 + 0 + . . . = 1, |||δ 2||| = 1, . . . , |||δ m||| = 1 |||x||| = |t1 | + |t2 | + . . . |tm | ≤ 1 |||x||| ≤ 1. ⇒x∈D ⇒A⊂D On the other hand let x ∈ D. ⇒ |||x||| ≤ 1 and x ∈ Φ. But x ∈ Φ ⇒ x is a finite sequence.
(2) (3)
⇒ x = (x1 , x2 , . . . xm , 0, 0, . . . ) = (x1 , 0, . . . ) + (0, x2 , . . . ) + . . . + (0, 0, . . . , xm , 0, 0, . . . ) = x1 (1, 0, 0, . . . ) + x2 (0, 1, 0 . . . ) + . . . + xm (0, . . . , 1, 0, . . . ) = x1 δ 1 + x2 δ 2 + . . . + xm δ m . where |||x1 ||| + . . . + |||xm ||| ≤ |||x||| ≤ 1. Hence |||x1 ||| + . . . + |||xm ||| ≤ 1. Thus, x = x1 δ 1 + x2 δ 2 . . . + xm δ m with ||x1 || + . . . ||xm || ≤ 1 ⇒ x ∈ A ⇒D⊂A (4) From (3) and (4), A = D. ⇒ E is a determining set of . Theorem 2: {δ k } is a Schauder basis for .
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Some vector sequence spaces-II
Proof: n
xk δ k
= x1 (e, 0, 0, . . . ) + x2 (0, e, 0, . . . ) + . . . + xn (0, . . . , e, 0, . . . )
k=1
= (x1 , x2 , . . . , xn , 0, . . . ) = x[n]
(5)
[n]
⇒ |||x − x ||| = |||(0, 0, . . . , 0, xn+1 , xn+2 , 0, . . . )||| ∞ xk → 0 as n → ∞ = k≥n+1
⇒ |||x − Also if |||x −
n k=1 n
xk δ k ||| → 0as n → ∞.
(6)
yk δ k ||| → 0 as n → ∞
k=1
x =
xk δ k =
yk δ k
⇒ (x1 , x2 , . . . ) = (y1 , y2, . . . ) ⇒ xk = yk ∀k ⇒ {xk } is a unique.
(7)
From (6) and (7) it follows that {δ k } is a Schauder basis for . Theorem 3: Let A = (ank ), (n, k = 1, 2, . . . ) be an infinite matrix. Then A ∈ ( : ∞ ) ⇔ sup ||ank || < ∞ (n,k)
Proof: E = (δ 1 , δ 2 , . . . ) is a determining set for . Also E = (δ 1 , δ 2 , . . . ) is a Schauder basis for , Hence has AK. We know that ∞ is a BK space with the norm |||x||| = sup(k) ||xk ||. A ∈ ( : ∞ ) ⇔ 1. The columns of A are in ∞ 2. A(E) is a bounded set in ∞ W e shall calculate A(E). W e have A(E) = A{(δ 1 , δ 2 , . . . )} ∞ ank xk with xk = 1, xi = 0 & x = k An (x) = k=1
= (ank )∞ k=1 ∀n
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K. Chandrasekhara Rao, S. Tamilselvan and K. Vairamanickam
But A(E) is bounded ⇔ |||(ank ||| is bounded in ∞ ⇔ sup ||ank || < ∞ (n,k)
Theorem 4:A ∈ ( : ) ⇔ sup(n) Proof :
∞
k=1 ||ank ||
A(E) is bounded in ⇔
∞ k=1
⇔ sup
< ∞.
||ank || is bounded unifomly in ∞
(n) k=1
||ank || < ∞.
Theorem 5: ∗ = ∞ Proof : x ∈ ⇒ x = xk δ k ⇒ f (x) = xk yk , yk = f (δ k ) Also ||yn || = ||f (δ n)|| ≤ ||f || ||δ n || = ||f || ⇒ y ∈ ∞ Also |||y||| = sup ||yn || ≤ ||f ||
(8)
(n)
Define T : ∗ → ∞ by T (f ) = y From (8) |||T (f )||| ≤ ||f ||
(9)
Give y ∈ ∞ , define f on by f (x) =
xn yn ⇒ f is linear. Also ||f (x)|| ≤ ||xn yn || ≤ sup ||yn || ||xn || (n) ≤
sup |yn | |||x||| (n)
⇒ f is bounded and hence continuous ⇒ f ∈ ∗ with T (f ) = y ⇒ T is subjective Also |f (x)| ≤ ||f || |||x||| where ||f || = sup |f (x)| ≤ |||y(f )||| |||T (f )||| ||x||≤1
(10)
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Some vector sequence spaces-II
From (9) and (10) |||T (f )||| = ||f || ⇒ T is a linear Isometry of ∗ on to ∞ ⇒ ∗ = ∞ . Theorem 6: c∗0 = Proof : x = xn δ n ⇒ f (x) = xn yn yn = f (δ n ) ∀f ∈ c∗0 . Let x(n) = nk=1 zk δ k Where zk = ||yykk || , 1 ≤ k ≤ n But then ||x(n) || ≤ 1 From (11) f (x(n) ) = nk=1 ||yk || ⇒ nk=1 ||yk || = |f (x(n) )| ≤ ||f || ||x(n) || ≤ ||f ||
(11)
n Hence (sn ), sn = k=1 ||yk || is a Monotonically increasing sequence bounded above ⇒ (sn ) converges to its supremum, letting n → ∞ ∞
yn ≤ f and so y ∈ l ⇒ ||y|| ≤ ||f ||
(12)
n=1
Define T : c∗0 → by T (f ) = y = yn From(12) ||T (f )|| ≤ ||f ||
(13)
Now for any given y ∈ ||xn yn || Converges ∀x ∈ c0 Define f on c0 by (11).Then |f (x)| ≤ ||xn yn || ≤ ( ||yn||)||x|| ⇒ f ∈ c∗n with T (f ) = y ⇒ T is a Adjective. Also ||f || ≤ ||yn || = ||y|| = ||T (f )|| By(13) and (14) T is a linear bijective such that ||T (f )|| = ||f || ⇒ c∗0 = l. Definition: bv0 = {x : ∞ k=1 ||xk − xk+1 || < ∞}. bv0 is a BK-Space ∞ with the norm |||x||| = k=1 ||xk − xk+1 ||
(14)
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K. Chandrasekhara Rao, S. Tamilselvan and K. Vairamanickam
Notation we write e(1) = (e, 0, 0, . . . ) e(2) = (e, e, 0, 0, . . . ) e(k) = (e, e, . . . , e, 0, 0, . . . )(k times) And so on. Note that e(1) = δ 1 , e(2) = δ 1 + δ 2 , e(3) = δ 1 + δ 2 + δ 3 Theorem 7: Let E = (e(1) , e(2) , . . . ).Then E is a determining set for bv0 . Proof : A = The absolute convex hull of E. D = (The closed unit ball in bv0 ) ∩ Φ Let X ∈ A ⇒ x = (t1 e(1) + t2 e(2) + . . . + tm e(m) ) with ||t1 || + ||t2 || + . . . + ||tm || ≤ 1, and also ti ∈ X x = (t1 + t2 + . . . + tm , t2 + t3 . . . + tm , . . . , tm , 0, 0, . . . ) ⇒x∈φ
(15)
And |||x||| ≤ ||T1 || ||e(1) || + ||t2 || ||e2 || + . . . + ||tm || ||em || where ||e(1) || = ||e − 0|| + ||0 − 0|| + . . . = ||e|| = 1 ||e
(2)
= 1, . . . , ||e(m) || = 1
⇒ |||x||| ≤ ||t1 || + . . . + ||tm || ≤ 1
From(15) and (16)X ∈ D. Hence A ⊂ D.
(16
Some vector sequence spaces-II
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Let X ∈ D ⇒ |||x||| ≤ 1andx ∈ φ ⇒ x = (x1 , x2 , . . . , xm , 0, 0, . . . ) = (x1 , 0, . . . ) + (0, x2 , 0, . . . ) + (0, . . . , xm , 0, . . . ) = x1 (e, 0, 0, . . . ) + x2 (0, e, 0, . . . ) + . . . + xm (0, 0, . . . , 0, e, 0, . . . ) = x1 e(1) + x2 (e(2) − e(1) ) + x3 (e(3) − e(2) ) + . . . + xm (e(m) − e(m−1) ) m (xk − xk+1 )e(k) = k=1
By Abel’s inequality m where xk − xk+1 ≤ |||x||| ≤ 1 k=1
⇒x∈A ⇒D⊂A Thus D = A ⇒ E is a determining set for bv0 Theorem 8: Let A = (ank ) n, k = 1, 2, . . . be an infinite matrix with ank ∈ X then A ∈ (bv0 : ∞ ) ⇔ sup(n,k) ||a(n1) + a(n2) + . . . + a(nk) || < ∞ Proof : We know that E = (e(1) , e(2) , . . . ) is a determining set for bv0 . By the lemma A ∈ (bv0 : ∞ ) ⇔ 1. The columns of A are in ∞ ,and 2. A(E) is a bounded set in l∞ Then A(E) = (an1 + an2 + · · · + ank )∞ n,k=1 Since A(E) is bounded in ∞ We have ⇒ sup(n,k) ||a(n1) + a(n2) + . . . + a(nk) || < ∞ Thus A ∈ (bv0 : ∞ ) ⇔ sup ||an1 + an2 + . . . + ank || < ∞
References [1] Chandrasekhara Rao. K, and Balasubramanian, K. Matrix tranformations involving vector sequence spaces, Int. Journal of Math. Analysis, Vol. 2 (2008) No. 26, 1269-1274. [2] Chandrsekhara Rao. K, Tamilselvan. S and Karunakaran. V, Some Vector Sequence spaces, Acta Ciencia Indica, Vol. XXXIV M, No. 3: (2008), 1067-1071
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K. Chandrasekhara Rao, S. Tamilselvan and K. Vairamanickam
[3] Wilansky. A, Summability through Functional Analysis, North-Holland, Amsterdam, 1984. Received: March, 2009
