By Charles Swartz
In line with an introductory, graduate-level direction given through Swartz at New Mexico nation U., this textbook, written for college kids with a reasonable wisdom of element set topology and integration concept, explains the foundations and theories of sensible research and their functions, displaying the interpla
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Example text
Let m be the largest integer such that q(xi) 5 a/2 [if this i=1 inequality fails to hold for m = 1, set m = 0 and ignore this inequality]. Chapter 3 33 m+1 Then 0 5 m < n and n q(xi) > a/2 so q(xi) <_ a/2. The sums on i =1 i=m+2 the left hand side of the following inequalities have fewer than n terms or are 0 so the induction hypothesis gives m m q( xi) 5 2 q(xi) 5 a i=1 i=1 n q( n q(xi) 5 a xi) 5 2 i=m+2 i=m+2 q(xm+1) 5 a. These inequalities and the hypothesis give the result. Theorem 3 (Kakutani).
Ii) T is continuous at 0. Proof: Assume (ii) holds and let x e E. Let V be a neighborhood of Tx. Then -Tx + V is a neighborhood of 0 so 3 a neighborhood U of 0 in X such that TU c -Tx + V. Thus, x + U is a neighborhood of x with T(x + U) c V, and (i) holds. 43 Linear Operators and Linear Functionais 44 Definition 2. A linear map between TVS is said to be bounded if it carries bounded sets to bounded sets. Proposition 3. Let E, F be TVS and T : E -4F sequentially continuous. Then T is bounded. Proof: Let B c E be bounded.
8). Example 20. Let S be a compact Hausdorff space. C(S) is the subspace of B(S) consisting of all continuous functions. We assume that C(S) is equipped with the sup-norm; C(S) is a B-space under the sup-norm since convergence in the sup-norm is just uniform convergence on S. Example 21. Let (S, L, µ) be a finite measure space and let L0(µ) be the Chapter 2 23 vector space of all real-valued 1-measurable functions defined on S. Then f t = JS 1 +f in I I dµ defines a quasi-norm on L°(p) such that convergence is exactly convergence in p-measure ([TL], p.