Let X be a finite dimensional normed space (a) and A :X→X is a linear map. Show that A is compact. Is this result true if X is infinite dimensional and A is a continuous linear map ? Justify.

Q: Let Y be a vector subspace of a normed space X. Show that the closure of Y is also a vector…

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Q: Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If…

A: Given let V and W  be the vector space and T and U be the non-zero linear transformation from V to…

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: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

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Q: Let V be a finite dimensional, complex inner product space and let U be a subspace of V. Suppose v e…

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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: 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: If X is the subspace of `∞ consisting of all sequences of zeros and ones, what is the induced metric…

A: We have given that ,  X is the subspace of l∞ consisting of all sequences of zeros and ones. We need…

Q: Every continuous function from normed space X into normed space Y is bounded linear operator True…

A: True

Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: Let V be a vector space over R and W be an inner product space over R with inner product (, ),. If T…

A: To prove this, prove that it is an inner product assuming T as one-one. Now prove the reverse order…

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: Let W be a finite-dimensional subspace of an inner product space V , and let E be the orthogonal…

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Q: Let V = {(x, y): x 20) with standard operation on R2. Is V a vector space?

A: Given set is  V={(x,y):x=>0 }

Q: Let U, V, W be finite dimensional vector spaces over F, and S : U → V,T:V → W linear…

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Q: Let X be a topolegical space be a and let a Continuous map. Define the Set Y=3x€ X : flx) = x} X is…

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Q: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

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

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Q: Let V be an inner product space, and let W be a finite-dimensional subspace of V. If x $ W, prove…

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Q: Let X be a normed space, Xn, Yn EX such that Xnx, Yny, then 1-Xn + Yn → 4- ||xn-ynll→→llx - yll.…

A: Given that X is a normed space, xn, yn ∈X such that xn→x, yn→y. To find1. xn+ yn→x+y 2. λxn→λx ∀λ∈F…

Q: Let V ((x,y): x 20) with standard operation on R2. Is V a vector space?

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Q: Let X be a normed space then 1- ||0|| = 0 2-I1-xl = |지l, Vx E X 3- ||x - yl| = |ly - x|,Vx, y E X…

A: As per the policy, we are solving first three subparts. Please resubmit the question and specify…

Q: Show that with the subspace topology Uy on Y, the inclusion map : Y → X is continuous.

A: Given that, UY is subspace topology on Y. Suppose (X, Tx) be topological space and Y is subset of X.…

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

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Q: If X a normed space and y is a proper proper Linear subspace of X. show that y has empty interior.

A: We proved by contradiction. Suppose Y has a nonempty interior then Y contains a open ball. Then we…

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: Exercise 9.8. Let X be a normed space, and let X = (X*)* be the bidual (or double %3D dual) of X.…

A: Given:  X is a normed space and X** = X** is the bidual (or double dual ) of X. Also, the map JX : X…

Q: Let V = {(x, y): x 20} with standard operation on R². Is V a vector %3D space?

A: Our objective is to conclude V is a vector space or not.

Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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Q: Let V = {(x, y): x 20} with standard operation on R2. Is V a vector space?

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

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Q: Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If…

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Q: Suppose V is a 4-dimensional vector space, and that T is a linear operator on V with minimal…

A: Dimension of Ker T

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

A: Given Let W be a finite-dimensional subspace of an inner product space V.

Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = <x, y>z for all x…

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Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

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 V be a finite-dimensional vector space and let U be a subspace of V. Show that if Lim(U) =…

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Q: Theorem 2: Let X be a finite dimensional normed space and let r > 0. Then, the closed unit ball B[0;…

A: Theorems and definitions we are going to use: 1. Definition: (X,|| . ||) is said to be bounded if…

Q: (a) Let X and Y be normed spaces and F :X→Y be a continuous linear map. Show that F is closed . Is…

A: Given that,                   X and Y are normed spaces and F : X → Y is a continuous linear map. We…

Q: Let V be an inner product space and T: V →V be a linear map. Suppose ||T (r) || = ||r||. Prove that…

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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: Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || × |lx + || y lly and + || y ily…

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Q: Define A : L3/2[–1, 1] → C by Prove that A is a bounded linear map and compute its norm.

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Q: Let T: U→V be a linear transformation from a vector space U (F) into a vector space V(F) and U(F) be…

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

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