Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a linear functional fo e X1 such that fo(xo) =d(x y), fo(Y) =0 and ||foll =1.

Q: If V = R is a vector space and let Hbe a subset of V and is defined as H={(a,b,c): a² +b² = 0,c>0}.…

A: To show that H is not a subspace of a vector

Q: Prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and, whenever a ∈…

A: We need to prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and,…

Q: Let S = { (x, y, z, w) E R*; x – 2y + w = 0 and 2x + z – w = 0 } . - Prove that S is a subspace by…

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Q: Consider L = I – A with A : X → X is a compact linear operator in a normed space X. Show that the…

A: The given problem is related with functional analysis. Given that L = I - A , where A : X → X is a…

Q: Let W1 and W2 be subspaces of a vector space V having dimensions m and n, respectively, where m > n.…

A: Given, dimension of W1=mdimension of W2=n such that m≥n   We know, W1∩W2 is a subspace of W1W1∩W2 is…

Q: Let W and U be subspaces of a vector space V. Suppose that for every v € V there is a unique pair…

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Q: Suppose X is a topological space whose topology is coherent with a family B of subspaces.  Prove…

A: Given X is a topological space, whose topology is coherent with a family B of subspace.Using the…

Q: b) Suppose that V is finite dimensional and that T E L(V, W). Prove that there exists a subspace U…

A: In a finite-dimensional Vector space, every subspace is finite-dimensional.

Q: Let X be a normed space and Y C X a linear subspace. c) Show that Y is dense in X, if and only if…

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Q: (b) Let f be a bounded linear functional on a subspace M of a normed space X and let X, E X, x, 4 M.…

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Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

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Q: Let W be a subspace of Rn. Prove that the intersection of W and W⊥ is {0}, where W⊥ is the subspace…

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Q: (a) Let X be a normed linear space and M be a closed linear subspace of X. Let x, e X be such that…

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Q: If S is closed Subspace of a normed Space (M, f), then prove that the function fo : R defined by f.…

A: If S is a closed subspace of a normed linear space M, f. Prove that the function fQ:MS→ℝ is defined…

Q: Let A be a fixed m × n matrix. Prove that the set W = {x ∈ Rn: Ax = 0} is a subspace of Rn.

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Q: et X = {(x,y, z): Vx² + y² + z² <1} be a subspace R° with the standard topology and 0 = (0,0,0).…

A: Given,               X=x, y, z:x2+y2+z2 ≤1  be a subspace of R3 with the standard topology and o=0,…

Q: Let V be a vector space over F, let T€ L(V), and let WC be a subspace invariant under T. Prove that…

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Q: Let V and W be vector spaces with subspaces V1 and W1, respectively. If T: V → W is linear, prove…

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Q: 4. Use the definition of a subspace to prove that the set V:= :, y, z E R, y+ : 0 is a subspace of…

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Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: Consider the following linear operator; T : R´ → R° given by T(x, y, z) = (-y, x + z, z). Find a 2-…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: If Sz is a subspace of R" of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: This staement is true.

Q: Suppose T E L(V) and U is a subspace of V. (a) Prove that if U null T, then U is invariant under T.…

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Q: Let V, W be vector spaces over F and let T:V → W be a linear map. Prove that if T is injective, then…

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Q: If Sz is a subspace of R* of dimension 3, then there cannot exist a subspace S2 of R* such that S, c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠S2 and S2≠R4.

Q: 17. Prove that a subset W of a vector space V is a subspace of V if and only if W + Ø, and, whenever…

A: We Know that Let V be a vector space over the field F and let W E V . Then W will be a subspace of V…

Q: Let W = (a + bx + cx2 + dx'| c- 3d = 0} be a subspace of P. Then the dimension of W is equal to O…

A: dim(W) = number of vector in the basis of W.

Q: Suppose U = {(x, y,x + y,x – y, 2x) e F° : x, y e F}. Find a subspace W of F such that F5 = U + W.

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Q: 1) Suppose that V is a finite dimensional vector space over R. Prove that every T E L(V) has an…

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Q: if it was M, N,L Subspaces of the vector space v on the field F Prove that (Mn N) + (M n L) c M n (N…

A: Given that M, N, L are subspace of a vector space V over the field F. We have to prove that M∩N +…

Q: Let W = {(x, y, 1) | x, y, E R}. Is W a subspace of R3?

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Q: Prove that X is completely regular space if and only if it is homeomorphic to a subspace of a…

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Q: Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying B n W = Ø.

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Q: Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V,…

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Q: Let A C X. Prove that if d is a metric for the topology on X, then d[(A × A) is a metric for the…

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Q: Let X = c, the space of all convergent sequenes and M = {1= (1,) such that In = 0 %3D then M is a…

A: True

Q: Suppose that V is a vector space with basis B= {bi | i E I} and S is a subspace of V. Let {B1,... ,…

A: Given that V is a vector space with basis B=bi|i∈I and S is a subspace of V. Let B1,B2,B3,…,Bk be a…

Q: Let β be a basis for a subspace W of an inner product space V, and let z ∈V. Prove that z∈W⊥if and…

A: Let β be the basis for a subspace W of an inner product V, and let z € V.It is required to prove…

Q: Let M be a proper closed subspace of a Hilbert space Hand let xo e H, xo £ M . Find a bounded linear…

A: We prove by using Hahn Banach Theorem.

Q: Please solve this question. Thank you

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Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: Suppose W1 and W2 are subspaces of a vector space, V. Define W as follows: W = {w1 + W2 | W1 E W1…

A: As W1 is a subspace of V , hence (a) 0∈W1 (b) αw1+βw1'∈W1 for every w1 , w1'∈W1 As W2 is a subspace…

Q: Prove that S = {(x, y, z) E R³: 2x + 3y+z = 0} is a subspace of the vector space R3.

A: Explanation of the answer is as follows

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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 W, and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Given that  W1  and W2 are the sub spaces of a finite dimensional vector space V.

Q: Let W be a 3- dimensional subspace of R*, then Dim ( WnW) = Оз O 1 O None of these

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Q: Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a…

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Q: Let X be a topological space, and let Y C X have the subspace topology. Prove that C C Y is closed…

A: