1. Let V be an inner product space. Suppose f,g E V such that f - g Lg (a) Prove that 1l12= ||f- g|2 + ||9||2.

Q: Construct bijective maps (a) from [0, 1) to (0, 2]. (b) from R − {3} to R − {1} (c) from Z to Z −…

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Q: Define the closure of a sot I in a motric spack. How is the boundry it a sut I related to it clogwai

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Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: The linear operator L : R² R² given by L (²) 6x - a) (55x + y) = -1 Which of the following gives a…

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Q: Show that R is not compact.

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Q: Let T: P₂ P₂ be a mapping defined by T(a + a₁x + a₂x²) = ao + a₁ (x + 1) + a₂(x + 1)². a) Determine…

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Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

Q: A: pp. A: px p. . A: pp(3)x3 + p(1)x.

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: 2c

A: The answer has been provided.Explanation:

Q: Let R be the relation on Z given by R= {(a,b) : a < b}. %3D (a) Find R-1. (b) Find R which is Z x Z…

A: relation on Z.

Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: 1. a. b. Let V be the set of real-valued functions that are defined at each x in the interval (-∞,…

A: Please see the below pictures for detailed solution.

Q: Let x(bar) = (3,-5,2,1,-2) element of R5. Find the norm of x(bar) using the usual norm on R5

A: The norm of x is the distance between a point x to the origin.

Q: Exercise 3. Prove that a relation R on A is antisymmetric if and only if RNR-' C {(a, a) | a E A

A: We know that , A relation R on set A is antisymmetric  if either x , y∉ R  or (y , x) ∉ R whenever…

Q: Let X₁ = [1 3 −1 1]²‚×₂ = [3 −1 3 1]′, and x3 = [-1 2 −3 1]. Is b = T an element of span{X₁, X₂,…

A: No, we can write it.

Q: Prove that ||u+v||2 + ||u-v||2= 2||u||+2||v|| for any u and v in an inner product space V

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Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

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Q: Assume that V is a finite dimensional real inner product space and T € L(V) and self- adjoint. Show…

A: Suppose that    is  a  finite  dimensional real  inner product space  and   and self-adjoint.We…

Q: #1. Find an explicit description of the null space of A by writing NulA as a span of vectors. 1350…

A: We have to find the null space of the matrices.

Q: 2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.

A: Let ℙ denote the vector space of all polynomials on ℝ .We have to show that ℙ is infinite…