Use Alaoglu's theorem to show that if X is a Banach space, there is some compact...
Please show the steps thoroughly, thank you so much!
Let X be a compact Hausdorff space and Y a subset of X. Let J/ be the ideal of functions in C(X) vanishing on Y. In general, Amay not be isomorphic to C(Y). Evaluate the following statement: If Y is closed in X, then Ais isomorphic to C(Y). If you answer true, is this isomorphism also isometric?
Let X be a compact Hausdorff space and Y a subset of X. Let...
1.5.7 Prove the following separately Theorem 1.5.10. Let (X,d) be a metric space. (a) IfY is a compact subset of X, and Z C Y, then Z is compact if and only if Z is closed (b) IfY. Y are a finite collection of compact subsets of X, then their union Y1 U...UYn is also compact. (c) Every finite subset of X (including the empty set) is compact.
Jet f be continuons one to one m compact metric space X onto a metric space Y. Prove that f'Y ~ X is continuoms (Hint: use this let X and Y e metric space, and let f be function from X to Y which is one to one and onto then the following three statments are equivalent. frs open, f is closed, f is continuous.
3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if
3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if
Topology
C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
Exercise 29: Let FCK for some F closed and K compact. Use the definition of a compact set in terms of covers to show that F is compact.
Carefully and rigorously prove the following.
Let X be a metric space. Show X is compact if and only if every sequence contains a convergent subse- quence. Hint for (): Argue by contradiction. If there was a sequence with no convergent subsequence, use that sequence to construct an open cover of X, such that every set in the cover contains only a finite number of elements of the sequence. Then use compactness to get a contradiction. Hint for (): Let...
Assume that Kis a non-empty closed set In Banach space V and that T:K → K. Prove that the iteration method xn+1 = T(x") n = 0,1,2 ... converges (prove that || x— x|| →0, as n →0)
(a) By the Heine-Borel Theorem, show that R2 is not compact and
the
sphere
S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1}
is compact in R3.
(b) Show that R2 and S2 is not homeomorphic. (i.e. no continuous
bi-
jective function f between R2 and S2 such that the inverse function
f−1 is continuous).
Question 1. (2 marks) (a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere is compact in R3. (b) Show that R2 and S2...
Problem II. i) Let Tı and T2 be two topologies on the same space X. Suppose that T2 is finer than η. If (X,n) is compact, does it follow that (X,2) is compact? Conversely, if (X, T2) is compact, does it follow that (X, Ti) is compact? la. ii) Let Y C X be equipped with the subspace topology. Show that Y is compact in the subspace topology if and only if any cover of Y with open sets in...