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AMATH/PMATH 331: Applied Real Analysis Fall 2022 Assignment 7
发布时间:2022-11-16
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AMATH/PMATH 331: Applied Real Analysis
Fall 2022
Assignment 7
Due at 11:59pm on Tuesday 15 Nov 2022
1. [10 points] (a) Show that Q is not connected.
(b) Let f be a continuous function from the unit circle S1 = {(x,y) | e(x,y)e = 1) in R2 to R. Show that f cannot be one-to-one.
2. [10 points] Let c(X, Y) be the space of linear transformations between two normed vector spaces X and Y. Recall the operator norm eAe = sup {eAxe | x l X, exe = 1), for A l c(X, Y). (We are abusing notation here, using e e to denote all three norms.)
(a) Show that the operator norm can be written as eAe = sup { | | x l X, x ≠ 0 }. (b) Show that eAxe s eAeexe for all x l X (assuming for simplicity that eAe is finite).
(c) When X is finite dimensional, we always have eAe < ”. Using this assumption, prove that the operator norm is a norm on the vector space c(X, Y).
3. [10 points] The Parallelogram Law states that
eu + ve + eu − ve22 = 2eue + 2eve22
for all u, v in a normed vector space V.
(a) Prove that the Parallelogram Law holds in an inner product space V.
(b) Prove that the normed vector spaces (Rn, e e1 ) and (Rn, e e” ) do not satisfy the Paral-
lelogram Law.