(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. Prove that f has an extension g to N =span M U{x,} such that g = S.

Q: 3. Show that the set W = {(x, 0) : x e R} is a subspace of R?.

A: In order to show W is a subspace of R2 , we need to prove W is closed under addition. For any two…

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A: Here we have given that X1 be a closed subspace of a normed space X. So, the quotient space XX1 is…

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Q: Let Z be a proper subspace of an n-dimensional vector space X, and let xo € X-Z. Show that there is…

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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 (X, F) be a space. Prove that a subspace of a subspace is a subspace of (X, F).

A: Here given X, I is a topological space.

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…

A:

Q: Let T and T' be topologies on a set X such that T' is strictly finer than T. If Y is a subset of X,…

A: Definition:  Let Τ and Τ' be topologies on a set X. If Τ' is strictly finer than Τ, then Τ⊂Τ'.…

Q: Let T be a linear operator on a finite-dimensional vector space V. Prove that the following…

A: An invariant subspace W of T has the property, such that vectors v belongs to W.

Q: Let W be a finite-dimensional subspace of an inner product space V. Show that if T is the orthogonal…

A: Given:W be a finite-dimensional subspace of an inner product space VTo prove:If T is the orthogonal…

Q: If A is a normed subspace of E, proof that bd(A) is a subspace of X.

A: Please check the answer in the next step

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…

A:

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 Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY…

A:

Q: 2. Let V, W be two vector spaces and let F: V H W be a linear map. Let U be the subset of V…

A: Given problem is :

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 M be a closed linear subspace of a Banach space X. Prove that the quotient space X/M, with the…

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Q: Let W1 and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Consider W1 and W2 be the subspaces of finite dimensional vector spaces V.

Q: ) Let T' be a linear operator on a vector space V and let W be a T-invariant subspace of V. Let n :…

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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: 6. Let W be a finite dimensional subspace of an inner product space V. For each w e W, there exists…

A:

Q: Let V be a real vector space, EC V a subspace, and T : V → a linear transformation satisfying T(E) C…

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Q: 3. Let V be a finite-dimensional vector space, and let W1 be any subspace of V. Prove that there is…

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Q: In this set of exercises, H always denotes a Hilbert space. 1 If M is a closed subspace of H, prove…

A: Given: M is a closed subspace of a Hilbert space H.   To prove: M=M⊥⊥   To determine: If there is a…

Q: Let V be a finite dimensional vector space and W CV a subspace. Prove that W has a complimentary…

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Q: Let W C R be the subspace spanned by vị = and v2 = . Compute the prthogonal projection of w = 3 onto…

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Q: Suppose V is finite-dimensional, with dim V = n > 1. Prove that there exist 1-dimensional subspaces…

A: Let us consider that (v1, v2, ..., vn) be a basis for V, and let Ui= span(vi), for i ∈ {1, 2, ...,…

Q: (a) What is the subspace topology R on Y induced by T? (b) List all of the R-closed subsets of Y.…

A: Subspace topology B on Y induced by I :                   B=(Y,IH)

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

A: Let W1  and W2 be the subspaces of finite dimensional vector space V which satisfy dimW1=m  and…

Q: 4. Consider C" as a real vector space. It has a subspace U := C x {0}"-1. What is the dimension of…

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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 T and T' be topologies on a set X such that T' is strictly finer than T. If Y is a subset of X,…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs…

Q: Prove that a subset W of a vector space V is a subspace of V if and only if 0 EW and ax+ y E W…

A: First proving the first part: If W is the subset of V and also the subspace of V Claim  : 0 W and…

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: Let M be the set of all vectors (or polynomials) x in p (over R) for which x(t)=x(-t) - even…

A:

Q: let P5 be the standard vector space for polynomials of degree <=5 and U be of the set of all…

A: Let P5(x) be the standard vector space for polynomials of degree≤5. and U be the set of all…

Q: Let T be a linear operator on a finite dimensional vector space V, and Let W1,W2,.....Wk be T…

A: Assume that T is diagonalizable. Then there exists an ordered basis of V, v1,v2,...vk  such that…

Q: 10. Let C' (R) be the set of all continuously differentiable functions. Show that W = {f : f' - Af =…

A:

Q: Let M be the set of all vectors (or polynomials) x in p (over R) for which x(t)=x(-t) holds…

A: Given M be the set of all vectors x in PR for which xt=x-t. A non empty subset M of a vector space…

Q: Let T be a linear operator on a finite-dimensional vector space V, and suppose that W is a…

A: Let p(x) be the minimal polynomial of T, hence p(x) annihilates T, i.e.,

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

A:

Q: Let X be a normed linear space which is non-zero. Prove that X is a Banach space if and only if the…

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Q: Find a basis for orthonormal complement W- for a set W, a subspace in R* W = [2x, + x, + X + 2x, = 0…

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Q: on

A:

Q: 8. If Z is an (n−1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null…

A: Given: n-1 dimensional subspace To find: Scalar multiple

Q: Let W be a subspace of R". Set w+ = {x € R"|x™w = 0 for each w in W}. Show that W- is a subspace of…

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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.