Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

Q: If τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normal space…

A: τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normalspace or…

Q: 1. Prove that for every non-gero normed space X there is a non-zero bounded linear fmetional acting…

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Q: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

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Q: Prove: If S is compact, and ƒ is continuous on S, then ƒ takes a minimum value some- where in S.

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Q: Prove that L∞[a, b] is complete with respect to the norm || · ||∞.

A: The elements of L∞[a,b] are the essentially bounded measurable functions.

Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: Let V be an inner product space, and suppose that T: V→V is linear and that ||T(x)|| = ||x|| for all…

A: Given:

Q: 3.3.9. Given a normed space X, prove that the function p: X R defined by p(x) = ||x|| is continuous…

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Q: 4) Consider the collection of intervals on the real line: B = { (a, b) | a 1}. Is this a basis for…

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Q: For a normed space X and a closed convex subset A of X. Prove that (º )nÎ= Â. A

A: Let, be a normed space and be the closed convex subset of As, we know that norm on is a function…

Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: Let X be a discrete spaces then O X is homeomorphic to R if and. only if X is countable O X is…

A: I have provided a solution in step2

Q: Let X be a normed space. (a) If E₁ is open in X and E₂ C X, then E₁ + E₂ is open in X. Proof: (a)…

A: It is given that X be a normed space, E1 is open in X and E2 be any subset of X. We have to explain…

Q: Find all values of m such that a(r) = {X – 1 if x m is continuous everywhere.

A: Questions from real analysis.

Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: Let X= {A ⊆ ℝ| 1 is not in A and ℝ \ A is finite}. Prove that ℝ is a compact Hausdorff space with…

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Q: Let E₁ and E₂ be open dense subsets of a topological space X. (a) Show that E₁ E2 is dense in X. (b)…

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Q: Extend the linearly independent Set S={x+1, 2X²} to the space base P₂ (R)

A: We are given a linearly independent set S= {x+1, 2x2} We have to extend S to basis of P2(R).    Let…

Q: Let P denote the set of functions f in C([0,1]) with f(x) > 0 for all x. Then P is a closed subset…

A: Given that, P denotes the set of functions f in C0,1

Q: Let R be equipped with the standard topology. Define q : R? → R, q((x1, x2) : Jx1, if x1 2 x2 |X2,…

A: Question: The function q:ℝ2→ℝ defined by qx1,x2=x1, if x1≥x2x2, if x1&lt;x2 where ℝ2 and ℝ are…

Q: Let X be a normed space and A € BL(X) be of finite rank. Then σe (A) = σ₁(A) = o(A).

A: 1

Q: If x is a limit point of a subset A of a metric space X, then sh

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Q: Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an…

A: please see the next step for solution

Q: Let K be a compact subset of a metric space X, and assume that -R is continuous. Prove that f…

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Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: Let X be a Banach space and Y be any normed space and F : X → Y be linear with the following…

A: Let X be a Banach space and Y be any normed space and F : X→Y be linear with the following property:…

Q: (13) Show that | l is a norm on the space L [a,b].

A: Please check step 2 for the solution.

Q: Let X be a topological space. Construct a topological space Px and a continuous surjection 0: X→ Px…

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Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.

Q: Let X and Y be normed linear spaces and F:X-Y. If for every Cauchy sequence {x, in X, the sequence…

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Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: Given that, X is a discrete space. To be homeomorphic X has to be of same cardinality as ℝ and so X…

Q: Every T2 (Hausdorf f) space is T1 space. True O False O

A: Since T2 is a product preserving topological property. So T2 space is a T1 space.

Q: Let (R, d) be discrete metric space and A = N then A is complete and compact neither complete nor…

A: Given that (ℝ,d) is a discrete matric space and A=ℕ. then     d(x,y)=0 if x=y1  if x≠y A matric…

Q: Then ||-|| is a norm on R.

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Q: Let X be a normed linear space such that its dual X' is separable, then show that X is separable ?…

A: This is a problem of Functional Analysis.

Q: Prove that if g′ is bounded on (a, b) then g is uniformly continuous on (a, b).

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Q: 10. If A is 3 x 3 with rank A 2, show that the dimension of the null space of A is 1.

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Q: Let A be dense subset of (X, t)

A: Given: Let A be dense subset of x,Τ and let B be non empty open subset of X. To find: Fill in the…

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: 3.7.5. Given a finite-dimensional normed space X, prove the following state- ments. (a) X is…

A: Multipart :    Let X be finite dimensional normed space. (a) We need to show that X is separable. As…

Q: Use th properties of linear Space A upon t to prove s a) 0.X - o for all X EX b) a.x=0 =>(a=07v…

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Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is 1.

A:

Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…

Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

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Q: Let X be a discrete spaces then * X is never homeomorphic to R O X is homeomorphic to R if and only…

A: Fourth option is correct.