1. Let A CX be a metric subspace of a metric space X. Show that A is totally bounded if and only if for any e > 0, there is a finite set {x₁,...,n} CX such that n ACU Be(x₂) i=1

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: i. Let (B, || – || B) be a Banach space and let A C B be a subspace (as a vector space) of B. Recall…

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Q: = Vi O Let T be the linear operator on V that satisfies T(U(v;)) (1 ≤ i ≤ k) and T(w;) = 0 (1 ≤ i ≤…

A: In this solution, we will consider a linear operator U on a finite-dimensional inner product space V…

Q: 2.57 † Let S = functions on [a,b]. Show that if n e N and if ao, a1,..., an are not all 0, then it…

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Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: 1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show…

A: Given topological space X such that X contains an infinite closed discrete subspace. We now show…

Q: .: Remind that the set F of all functions from R to R is a vector space. Prove that the set E of…

A: We know that If W is a subset of a vector space V , then W is a subspace of V if and only if the…

Q: If A, B are subsets of the topological space X show that Ext(AUB) = Ext(A)n Ext(B).

A: Given: A and B are subsets of the topological space X. To Prove: ExtA∪B=ExtA∩ExtB

Q: a normed space, mapping such that ker(T) = {0}). Show that: defines a norm on E. any vector space…

A: Answer given below.

Q: 6. Let V be a vector space with subspaces U and W. Show that the set UnW = {x|x EU and xEW} is a…

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Q: 3. Determine, with proof, whether each set is a subspace of the given vector space {a, + a1x + a2x²…

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Q: 1. Let X be a metric space and A C X be a metric subspace. (a) If A is closed in X, prove or…

A: be a metric space and  be a metric subspace.

Q: 2.53 Let B[a, b] denote the set of all bounded functions on [a, b). Is Bla, b] a vector space? Prove…

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