Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator norm if and only if Y is a Banach space.

Q: Use Hahn-Banach Theorem to show that a normed linear space X is finite dimensional if its dual X' is…

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Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ × V₂, W) of bilinear maps is a…

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Q: Prove (Lo (x),|- ||∞) is a normed space

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Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: _I. Let H and K be Hilbert spaces and A E B(H, K). Show that A is compact if = and only if A* A is…

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Q: Бxї Zet (x, г) бе space, then Co-countable тебя xis not Тз-брасе

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Q: show that the discrete metric on u Vector space X+3 of eannot be obtained from a norm.

A: To Show: Show that the discrete metric on a vector space X ≠ 0 cannot be obtained from a norm.

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: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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Q: . Exercise 1.2.10 Let V₁ and V₂ be normed spaces with norms || ||1 and ||||2. Recall that the…

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Q: If X/M ={ [x], xe Xx} show that X/M is a linear space, Where [x] is equivalence class

A: To show that X/M is a linear space, we need to verify that it satisfies the following axioms:…

Q: 3. Prove that, for a given n, all n-dimensional vector spaces with real scalars are isomorphic.

A: A vector space V will be isomorphic to other vector space W if there exists a linear transformation…

Q: Determine whether the vector spaces Ps(R) and M2,6 (R) are isomorphic.

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Q: (c) Prove that the adjoint of a compact operator is compact.

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Q: Prove that coo cannot be a Banach space under any norm.

A: To prove: c00 cannot be a Banach space under any norm.

Q: Let X & Y be normed spaces & x # 204. Then BL (X₂Y) is a Banach space in the operator norm if and…

A: Here BL(X,Y) (or, B(X,Y)), is the space of bounded linear operators from X to Y. First ,we show that…

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: By example prove that a Housdorff space with a countable basis need not be metrizable.

A: We know that a topological space is metrizable then it is regular. For example, Let ℝ be a set of…

Q: Prove that Co is a Banach space under the supremum norm.

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Q: Let U = ((1, 1, 0, 1), (0, 1, 2, 3), (2, 3, 2,−1)) and W = ((1, 3, 4, 4), (1, 1, 0, 2)) be sub-…

A: The given U=1, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, -1 and  W=1, 3, 4, 4, 1, 1, 0, 2 be a subspaces of ℝ4.…

Q: Let T be an invertible linearlo-perator on a finite-dimensionalvector spaceProve that T is…

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Q: 2. Determine whether the set R2 with the operations (x1, y₁) + (x2, Y2) = (x1x2, 9192) and c(x₁, y)…

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Q: he set V = A {(x, y) ∈ R² | x> 0 and y> 0 |} is a vector space

A: False

Q: Show that a bounded linear operator A on complex Hilbert space H is self adjoint if and only if is…

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Q: 18. Let V= {(a1, a2): a1, a2 € R}. For (a₁, a2), (b1,b2) EV and c € R, define (a1, a2) + (b₁,b2) =…

A: An operation vector addition ‘ + ‘ must satisfy the following conditions:Closure : If x and y are…

Q: Prove that any linear map between finite dimensional normed linear spaces is continuous.

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Q: Prove that a linear map on a normed vector space is bounded if and only if it is continuous.

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Q: Suppose V and W are finite-dimensional vector spaces. Prove that there is an injective linear…

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Q: Let A: X₁ ≤X → Y be a closed operator where X and Y are Banach spaces. Define ||x|| = ||x||+||Ax||,…

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Q: (b) Suppose -|, and ||-||2 are two equivalen norms on a vector space X. If (X, ||.||,) is complete,…

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Q: (b) Let X be a normed space and Y be a Banach space. Define the operator norm on the linear space…

A: (b) It is given that X is a normed space and Y is a Banach space. Firstly, define the BX,Y  as,…

Q: a Normed &pace and and CL(X,Y) denste, of bounded Let X,Y be a Suppose BL(X,Y) respectively, the…

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Q: Prove that a closed linear subspace of a reflexive space is reflexive.

A: This is a problem of functional analysis.

Q: prove that any finite dimensional normed space is reflexive.

A: Let us suppose that we have a finite-dimensional normed space, X For any natural number, n, let…

Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ x V₂, W) of bilinear maps is a…

A: Please check step 2 and 3 for solution.! We will verify axioms of vector space.

Q: Let V be an inner product space and T be a linear operator on V with || Tv||= ||v||  ∀ v ∈ V. Show…

A: Let V be an inner product space and T be a linear operator on V with Tv=v ∀v∈V.

Q: 4. Let W be a subspace of a vector space V. We define a relation: V₁ V₂ if V₁ V₂ € W. ~ 4a. Show…

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Q: be a n-dimensional normed space, then prove that its dual space is also n-dimensional.

A: Let X be a normed linear space. Let X* be its dual space with the usual dual norm ||T|| = sup{…

Q: Is it true that if two vector spaces V and W both have dimension 2022, then they must be isomorphic?…

A: Suppose V and W are two vector spaces. We know that , V and W are isomorphic to each other if there…

Q: Given the Euclidean inner product space (R³(R),+,,*), where for each x=(x1,x2,X3), y=(yı,y2,y3) ER³,…

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Q: Characterise all bounded linear functionals on a Hilbert space.

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