Problem 10: Let CR ([-1, 1]) denote the vector space of continuous real-valued func- tions on the interval [-1,1] with inner product given by (f.g) = [ f(x)g(x)dx. Let be the linear functional on CR ([-1, 1]) defined by y(f) = f(0). Show that there does not exist g € CR ([-1, 1]) such that y(f) = (f,g) for all fe CR ([-1, 1]).

Q: Suppose f and g are non-constant, differentiable, realvalued functions defined on (−∞, ∞).…

A: We plug x=0 and y=0

Q: (a) Prove the set (A(*)./a(*),fa(a)) is linearly independent. (b) Use Gram-Schmit process to find an…

A: Given,     f1(x)=3,  f2(x)=1-x, and f3(x)=x2+2x-1

Q: Let L(x) be the linearization of an arbitrary function f about c. Assume that f is concave down in…

A:

Q: Let the invertible function Evaluate f(x) = 3x³+x+3 1 = ["] † ^¹ (2²) dar dx (Hint: You may use the…

A:

Q: Give an interval over which the set of two functions fi(x) = x² and f2(x) = x |x| is linearly…

A: Given: The set of two function.f1(x)=x2f2(x)=xx

Q: 1) Let X be a complex vector space and f2: X R be a linear functional. Define f:X C by f(x) = f2(ix)…

A:

Q: What is the Banach-Steinhaus theorem and how does it relate to the concept of bounded linear…

A: A normed vector space is a vector space with a norm, which allows us to quantify the distance…

Q: Let V be the vector space of all polynomials in the variable x over R. Show that the mappings…

A:

Q: 3. Determine whether each of these functions from is one-to-one/onto. (a) f : R → R, f(x) = 3r. (b)…

A: A function f is one-to-one if every element of the range of g corresponds to exactly one element of…

Q: 4: Which of the following functions has gradient vector [4x₁x₂ 2x² 3x²]¹? A: f (X) = 2x{+2x3+x3x3 B:…

A:

Q: anction of z = x + iy

A:

Q: af Ꭷf f(x, y) — f(y, x) = (x − y) — (x, y) + (y − x) — (x², y") ду or some point (x*, y*) € R² b)…

A: aSince given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation…

Q: Determine whether cach of these functions from Z to Z is one-to-one. a) f(n)=n- I c) f(n)=n b) f(n)…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…

Q: Suppose f(z) = z+z-2i 15 Define f(i) in such a way that the new function is . z +i continuous of…

A:

Q: If yo <0 then the interval of validity for IVP: y = y? ,y(0) = yo is: Celect one: O a. None of the…

A:

Q: A Show that the function f: R² → R defined by f(x₁, x₂) = x² + 2x² + 3x₁x₂ + 4x₁ + 5x₂ + 6 is a…

A: Definition of Vector Quadratic Function : Given Function f: R2→R is a vector quadratic function if…

Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V × V → V…

A:

Q: 1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2). Let f,…

A: We can just multiply h and k and integrate  Then after simplifying we are getting answer

Q: 10. Let V be the vector space of functions spanned by B = {cos x – sin a, cos a + sin æ}. Find the…

A:

Q: Suppose f(z) = u(x, y) is a real valued function on an open set U which we are viewing as a…

A:

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

A:

Q: use Euler theorem to show that if f (x, y) is hamog of degree one than x + 0 8 y to fxx (x,y) fyy…

A:

Q: 2) Let X be normed space and xo + 0 be an element of X. Then using Hahn-Banach Theorem (Normed…

A: We will construct a functional satisfying given data

Q: a) Let f: R - [1, co), where f (x) = x2 + 1. Determine whether the function maps onto the given…

A:

Q: Show how a real

A: To Prove that how a real linear functional u on a complex linear normed space gives rise to a…

Q: f: R →R by f(x)= x^2 and V is a subset of the codomain. Find a set V such that V has only one…

A: We are given that f : R → R by f(x) = x2 and V is a subset of the codomain. We need to find the set…

Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

A:

Q: 5. Prove that for any vector x, we have (a) 1, <H<#H, (b)

A:

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: the degree of homogeneity, if there is one, for eac 1 X1,X2, X3) = - xf +x +x X2 X3 O gro finotion:…

A: Introduction: For a function f(x,y), the degree of homogeneity is an integer n such that,…

Q: 2) Find the indicated norms of the following linear functionals defined on RS b) f(x1,X2, X3, X4,X5)…

A: This question is about application of functional analysis

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: Example: Show that the function Az) = z² is analytic. z = x+iy S(2) = z

A:

Q: Consider the relation x2z2 = arctan (xyz) + 4, which defines z as an implicit function of x and y,…

A:

Q: Let f, g and h be R- R functions, f(x) = 2x - 1, g(x) = 3x+1, and h(x) = x +1. Compute the following…

A: I will solve all three subparts for you and provide detailed answers.