Compactness in Product Spaces

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1 Pure Mathematical Sciences, Vol. 1, 2012, no. 3, Compactness in Product Spaces WonSok Yoo Department of Applied Mathematics Kumoh National Institute of Technology Kumi , Korea Abstract We establish some basic facts for compactness in product spaces and then derive a series of important results in analysis and measure theory. Mathematics Subject Classification: 54B10 Keywords: Product topology, Compactness, Equicontinuity, Uniformly countably additive family of measures 1 Introduction Let Ω and X ω be a topological space for each ω Ω. Let ω Ω X ω be the product space with the product topology σω which is the topology of coordinatewise convergence. For each ω Ω, P ω : λ Ω X λ X ω is the projection such that P ω ((x λ ) λ Ω )=x ω, (x λ ) λ Ω λ Ω X λ. If each X ω is a topological vector space, then with the coordinatewise operations and the product topology σω, ω Ω X ω is a topological vector space. In this paper we would like to establish a basic proposition for compactness in the product space ω Ω X ω and then we derive a series of important results in analysis and measure theory. Observe that for ω Ω and S ω Ω X ω, P ω (S) ={x ω :(x λ ) λ Ω S}. 2 Main Results Lemma 2.1. If S ω Ω X ω such that P ω (S) is relatively compact in X ω for each ω Ω, then S is relatively compact in ω Ω X ω =( ω Ω X ω,σω).

2 108 WonSok Yoo Proof. By Tychonoff product theorem, ω Ω P ω(s) is compact. If (x λ ) λ Ω σω S, then there is a net ((x αλ ) λ Ω ) α (I, ) in S such that (x αλ ) λ Ω (x λ ) λ Ω, i.e., lim α x αλ = x λ for each λ Ω. Hence S ω Ω P ω(s) ω Ω P ω(s) since ω Ω P ω(s) is closed in ω Ω X ω ([2, p.100]). But ω Ω P ω(s) is compact and S is closed in the compact ω Ω P ω(s). Hence S is compact in ω Ω X ω. Lemma 2.2. If X ω is Hausdorff for each ω Ω and S is relatively compact in ω Ω X ω, then P ω (S) is relatively compact in X ω for each ω Ω. Proof. S is compact in ω Ω X ω and so P ω (S) is compact in X ω for each ω Ω. Since each X ω is Hausdorff, P ω (S) is closed so P ω (S) =P ω (S) for all ω Ω. Moreover, if (x λ ) λ Ω S, i.e., there is a net ((x αλ ) λ Ω ) α I in S such that lim α (x αλ ) λ Ω =(x λ ) λ Ω S, i.e., lim α x αλ = x λ for each λ Ω, then x ω = lim α P ω ((x αλ ) λ Ω ) P ω (S) for each ω Ω and so (x ω ) ω Ω ω Ω P ω(s), S ω Ω P ω(s). Then P ω (S) P ω (S) and P ω (S) P ω (S) =P ω (S) P ω (S), P ω (S) =P ω (S), ω Ω. Hence P ω (S) is compact in X ω for each ω Ω. Corollary 2.3. If X ω is Hausdorff for all ω Ω and S ω Ω X ω, then S is relatively compact in ω Ω X ω if and only if P ω (S) is relatively compact in X ω for each ω Ω. For topological space Y =(Y,τ) and A X Y, let { } A (X,τ) τ = z X : net (x α ) α I in A such that x α z. Theorem 2.4. Let Ω and X ω be a Hausdorff topological vector space for each ω Ω and F a vector subspace of ω Ω X ω, S F. Let σω be the product topology on ω Ω X ω. Then S is relatively compact in (F, σω) if and only if 1. S (Q ω Ω Xω,σΩ) F, i.e., S (Q ω Ω Xω,σΩ) = S (F,σΩ), and 2. P ω (S) is relatively compact in X ω for each ω Ω. Proof. Suppose that S (F,σΩ) is compact in (F, σω). With σω and the coordinatewise operations, ω Ω X ω and its vector subspace (F, σω) are topological vector spaces and so the compact set S (F,σΩ) is complete in (F, σω) ([1, p.75]). Let ((x αω ) ω Ω ) α I be a net in S such that lim α (x αω ) ω Ω = (x ω ) ω Ω in ( ω Ω X ω,σω). Since S F ω Ω X ω, the convergent net ((x αω ) ω Ω ) α I is Cauchy in (F, σω) and so lim α (x αω ) ω Ω =(z ω ) ω Ω S (F,σΩ) F. Since each X ω is Hausdorff, ω Ω X ω is Hausdorff ([4, p.92]) and so (x ω ) ω Ω =

3 Compactness in product spaces 109 lim α (x αω ) ω Ω =(z ω ) ω Ω. Hence, (x ω ) ω Ω =(z ω ) ω Ω F and so S (Q ω Ω Xω,σΩ) = S (F,σΩ) F. Then S (Q ω Ω Xω,σΩ) = S (F,σΩ) is compact in (F, σω) and so S (Q ω Ω Xω,σΩ) is compact in ( ω Ω X ω,σω). Thus, (2) holds by Lemma 2.2. Conversely, suppose that both (1) and (2) hold for S. By Lemma 2.1, S (Q ω Ω Xω,σΩ) is compact in ( ω Ω X ω,σω) but S (F,σΩ) = S (Q ω Ω Xω,σΩ) F so S (F,σΩ) is compact in (F, σω). For a topological space X and Ω, let X Ω be the family of mappings from ΩtoX, and letting X ω = X for each ω Ω, the product space ( ω Ω X ω,σω) is homeomorphic with (X Ω,σΩ) by the correspondence (f(ω)) ω Ω f for each f X Ω. Then the following special case of Lemma 2.1 and Lemma 2.2 is a basic proposition in general topology ([4, p.218,th.1]). Corollary 2.5. Let X be a topological space and Ω, S X Ω. If S[ω] ={f(ω) :f S} is relatively compact for each ω Ω, then S is relatively compact in (X Ω,σΩ) where σω is the topology of pointwise convergence on Ω. If, in addition, X is Hausdorff, then the converse implication also holds. Corollary 2.6. Let X be a Hausdorff topological vector space, Ω and F is a vector subspace of X Ω, S F. With the topology σω of pointwise convergence on Ω, S is relatively compact in (F, σω) if and only if 1. S (XΩ,σΩ) F, i.e., S (X Ω,σΩ) = S (F,σΩ), and 2. S[ω] ={f(ω) :f S} is relatively compact for each ω Ω. If X is locally convex and X = { x C X : x is linear and continuous }, the dual of X, then there is a beautiful duality theory for the pair (X, X ). However, it is possible that X = {0} even for some Fréchet spaces such as M(0, 1), L p (0, 1) with 0 <p<1 ([1, p.25]), etc. Hence every reasonable extension of X will be interesting. Let C(0) = { γ C C : lim t 0 γ(t) =γ(0) = 0, γ(t) t if t 1 }. For a topological vector space X, γ C(0) and U N(X), the family of neighborhoods of 0 X, let { X (U,γ) = f C X : f(0) = 0, fis continuous, for x X, u U and t C, t 1, } f(x + tu) =rf(x)+sf(u) where r 1 γ(t), s γ(t). The usual dual X X (U,γ) for all U N(X) and γ C(0). Especially, it is possible that X = {0} but X (U,γ) is a large family ([6, 8]). Then we have a proper improvement of the Alaoglu-Bourbaki theorem as follows. As usual, f α f in (X (U,γ),σX) means that f α (x) f(x) at each x X, i.e., σx is the weak topology on X (U,γ).

4 110 WonSok Yoo Theorem 2.7. Let X be a topological vector space and U N(X), γ C(0). IfS X (U,γ) is equicontinuous, then S is relatively compact in (X (U,γ),σX), and for every V N(X), the polar V = { f X (U,γ) : f(v) 1, v V } is compact in (X (U,γ),σX). Proof. Suppose S X (U,γ) and S is equicontinuous. It is easy to see that if x α x in X then lim α f(x α )=f(x) uniformly for f S and so S (C X,σX) X (U,σX), sup f S f(x) < + at each x X, i.e., for each x X, S[x] = {f(x) :f S} is relatively compact in C. By Corollary 2.6, S is relatively compact in (X (U,γ),σX). Let V N(X). Then V = { f X (U,γ) : f(x) 1, x V } is equicontinuous on X ([8, Corollary 3.6]) and V is closed in (X (U,γ),σX). Thus, V is compact in (X (U,γ),σX). There is a very important Vital-Hahn-Sakes-Graves-Ruess theorem ([3, 9]) says that if X is a locally convex space and S is a relatively compact subset in the measure space (ca(σ,x),σσ) with the topology σσ of pointwise convergence on the σ-algebra Σ, i.e., μ α μ means that μ α (A) μ(a) at each σσ A Σ, then S is uniformly countably additive, i.e., if {A j } Σsuch that A i A j = (i j), then lim n n j=1 μ(a j)=μ( j=1 A j) uniformly for μ S. In 2006, R. Zi, Y. Yang and C. Swartz (RYC) have improved this important result to the following RYC s theorem. Let X be a locally convex space and Σ a σ-algebra. If S is countably compact in (ca(σ,x),σσ), then S is uniformly countably additive ([7, Corollary 4.3]). Theorem 2.8. Let X be a complete Hausdorff locally convex space and Σ a σ-algebra. Then for S ca(σ,x), the following (I), (II) and (III) are equivalent. (I) S is relatively compact in (ca(σ,x),σσ). (II) S is uniformly countably additive and {μ(a) :μ S} is compact for each A Σ. (III) S (XΣ,σΣ) ca(σ,x) and {μ(a) :μ S} is compact for each A Σ. Proof. (I) (II). By RYC s theorem, S is uniformly countably additive. By Corollary 2.6, {μ(a) :μ S} is compact for each A Σ. (II) (III). Let A j Σ, A i A j = (i j). Let (μ α ) α I be a net in S such σσ that μ α μ X Σ. Since X is complete, μ( j=1 A j) = lim α μ α ( j=1 A j)= lim α lim n n j=1 μ α(a j ) = lim n lim n α j=1 μ α(a j ) = lim n n j=1 μ(a j)= j=1 μ(a j). Thus, μ ca(σ,x) and so S (XΣ,σΣ) ca(σ,x). (III) (I). Corollary 2.6.

5 Compactness in product spaces 111 Now the famous Bartle-Dunford-Schwartz-Nikodým theorem ([5, p.305]) can be improved to the following. Corollary 2.9. Let Σ be a σ-algebra. For S ca(σ, C), the following (i), (ii), (iii), (iv) and (v) are equivalent. (i) S is weakly sequentially compact ([5, p.67]). (ii) S is uniformly countably additive and sup μ S,A Σ μ(a) < +. (iii) S is relatively compact in (ca(σ, C), σσ). (iv) S (C Σ,σΣ) ca(σ, C) and supμ S μ(a) < + for each A Σ. (v) S is uniformly countably additive and sup μ S μ(a) < + for each A Σ. Proof. By Bartle-Dunford-Schwartz theorem ([5, p.305]), (i) and (ii) are equivalent. By Theorem 2.8, (iii) is equivalent to the following. (II ) S is uniformly countably additive and sup μ S μ(a) < + for each A Σ. By the Nikodým boundedness theorem ([5, p ]), (II ) is equivalent to (ii) and so (iii) (ii) holds. By Theorem 2.8, (iii) (iv) (v) holds. A series of important facts in the basic theory of locally convex spaces are convenient consequences of Theorem 2.4. Corollary Banach space X is reflexive if and only if the unit ball B = {x X : x 1} is weakly compact. Proof.. Iff B (C X,σX ) σx, then there is a net (xα ) α I in B such that x α f and so f(x ) = lim α x α (x ) x for all x X, i.e., f X = X, f 1, f B. Thus, B (C X,σX ) = B X. Since supx B x(x ) x < + for each x X, i.e., B[x ] is relatively compact in C for each x X, Theorem 2.4 shows that B is compact in (X, σx ).. By Theorem 2.4, B (C X,σX ) (C X and so B X,σX ) = B. Then the Goldstine theorem shows that X = X. Corollary A subset S of a locally convex space X is weakly compact if and only if S is bounded and complete in (X, σx ).

6 112 WonSok Yoo Proof.. S X C X. By Theorem 2.4 and the Mackey theorem ([1, p.109]), S is bounded in X. Since S is compact in (X, σx ), S is complete in (X, σx ) ([1, p.75]). σx. If (x α ) α I is a net such that x α f C X, then (x α ) α I is Cauchy in (X, σx σx ) and so x α f S. Hence S (C X,σX ) = S X and S is weakly compact by Theorem 2.4. Corollary Let X be locally convex and S X. Then S is weak compact if and only if S is pointwise bounded on X and S is complete in (X,σX). Proof. Theorem 2.4. For A X, A = {f X : f(x) 1, x A}. Corollary Let X be a locally convex space and S X. The following (a), (b) and (c) are equivalent. (a) S U is compact in (X,σX), U N(X). (b) S U is closed in (X,σX), U N(X). (c) If (f α ) α I is an equicontinuous net in S such that f α f S. σx f C X, then Proof. (a) (b). (X,σX) is Hausdorff and S U is compact in (X,σX) for each U N(X) and so (b) holds. (b) (c). [(f α ) α I ] N(X), (f α ) α I S [(f α ) α I ] σx and f α f. Then f S [(f α ) α I ] and so f S. (c) (a). Let U N(X). U U, U N(X) and hence U is equicontinuous. Let (f α ) α I be a net in S U σx such that f α f C X. By (c), f S. But f(u) = lim α f α (u) 1 for all u U and so f X, f S U, S U X,σX) (C = S U X. If x X and f S U, then 1 n 0 x U for some n 0 N and so f( 1 n 0 x) 1, f(x) n 0, {f(x) :f S U } is compact for each x X. Thus, (a) follows from Theorem 2.4. A topological vector space is BTB if each bounded set in X is totally bounded ([1, p.85]), e.g., R n, C N and the space D of text functions, etc. If X is complete BTB, then bounded sets in X are relatively compact. Theorem 2.4 can be used to show that we have BTB spaces as many as all sets. Corollary If X is a complete BTB space, then for every Ω the product space (X Ω,σΩ) is a BTB space.

7 Compactness in product spaces 113 Proof. Let S be a bounded set in (X Ω,σΩ). Then {g(ω) :g S} is totally bounded in X for each ω Ω and so {g(ω) :g S} is compact in X for each ω Ω. By Theorem 2.4 or Corollary 2.5, S is relatively compact in (X Ω,σΩ). Thus, (X Ω,σΩ) is BTB. Corollary Let X be a topological vector space and Y a complete Hausdorff BTB space, S L(X, Y )= { f Y X : f is linear and continuous }. If S is equicontinuous, then S (Y X,σX) is compact in (L(X, Y ),σx). Proof. Since S is equicontinuous, S (Y X,σX) L(X, Y ), i.e., S (Y X,σX) = S (L(X,Y ),σx). By the equicontinuity of S again, S is pointwise bounded but the range space Y is complete BTB and so {g(x) :g S} is compact for each x X. Then the desired follows from Theorem 2.4. ACKNOWLEDGEMENTS. This research was supported by Kumoh National Institute of Technology. References [1] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, [2] A. Wilansky, Topology for Analysis, John Wiley, [3] J. Diestel and J. J. J. Uhl, Vector Measures, Vol. 15 of Math. Surveys, Amer. Math. Soc., Providence, RI, [4] J. L. Kelley, General Topology, Springer-Verlag, [5] N. Dunford and J. Schwartz, Linear Operators I, Interscience, N. Y, [6] R. Zi, S. Zhong and L. Li, Demi-linear analysis I-basic principles, J. Korean Math. Soc., 46(3) (2009), [7] R. Zi, Y. Yang and C. Swartz, A general Orlicz-Pettis theorem, Stud. Sci. Math. Hung., 43(1) (2006), [8] R. Zi, A. Choi and S. Zhong, Demi-linear duality, J. Inequal. Appl., to appear. [9] W. H. Graves and W. Ruess, Compactness in spaces of vector-valued measures and a natural mackey topology for spaces of bounded measurable functions, Contemp. Math., 2 (1980), Received: February, 2012