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: Q3 Assume that (M,d) is a Compact metrie Space. Show that : (M,d) → (ya) is Continous bjective then…

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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: Let (X₁, || ||1), (X2, ||· ||2) be Banach spaces. On X₁ X X₂ define || (X1, X₂) || = || X1 ||1 + ||…

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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: Define a bijective map g : x2/3--->R.

A: Sol

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: Let V be an inner product space. Show that ||cu|| |c| ||u|| for any vector u and any scalar c.

A: Given:          V is an inner product space.   To show :            cu = c u for any vector u and…

Q: 2c

A: The answer has been provided.Explanation:

Q: if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis…

A: Consider a discrete metric space X,d where d is defined as: dp,q=0,p=q1,p≠q

Q: Q3 Assume that (M,d) is a Compact metric Space. Show that, f: (M,d) → (Y, 2) is Continous bjective…

A: f:(X, d) to (Y, d) is said to be homeomorphism if  1) f is continuous.  2) f is bijective.  3) f…

Q: Let R+ be the set of all non-negative real numbers equipped with the following operations: (i) u ⊕…

A: Let R+ be the set of all non-negative real numbers equipped with the following operations: (i) u ⊕ v…

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: is an Let V(F) be an inner product space. If 0 + o e V then prove that orthonormal set.

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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: a) Let V(R) be the inner product space and a, B E V Il a +B 12+| a - B 1?= 2(1| a |l² +I B I²) prove…

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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 f(x) = + 3x? + 4x. a) What is the kernel of f? b) Compute the minimum and the maximum of f.

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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: Check whether or not if L/K and K/F are Galois extensions, then L/F is a Galois extension.

A: Suppose that EF and KF are extensions of a common base field F. Say that a field homomorphism φ: E→K…

Q: Let X be a pre-Hilbert space then [(x, y)|

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Q: X2 - X1' Let L:R3 → R³ be a linear operator such that L ×2 ) =| X3 - X2 , then Ker(L) spanned by: X3…

A: we know that Ker(L) is solution of the system L(X)=0  Reveal sold the system and find the ker(L)…

Q: Let v= C[-1,1] What is the norm of the functionf(x) = - x in this space? O A) 2V5 O B) 2V10/5 OC)…

A: An inner product in the vector space of continuous functions in [-1, 1], denoted asV=C-1, 1, is…

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: Prove that (F,|.|) is Banach space where FE{C, R}

A: Definition required:  Let X be a metric space and let xn be a sequence of points in X. (a) Cauchy…

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: If T₁ is the restriction of T to W₁ and T2 is the restriction of T to W₂, where, T:V→V is a linear…

A: To prove that f(x) is the minimal polynomial of T₁ and g(x) is the minimal polynomial of T₂, we need…

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…

Q: Find ß so that {1, x, 1+ 6x²) on (-1,1) is orthogonal.

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Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

Q: Consider the operator T: 1 given by T(r1, 22,. ) = (0, r1, 22,. ). Is T a compact operator? Explain…

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