(a) Show (by calculation) that {u1(x), u2(x)} is an orthonormal set. (b) Find the projection p(x) of f(x) onto Span(u1(x), U2(x)).

Q: Show that A is invertible if and only if AT is invertible.

A: Concept: Where the operator, operating on a function, produces a constant times the function, is…

Q: 4. Suppose that T : R* → R² is a linear map defined by T (x1, x2, x3, x4) = (2x1 x2 + x3, -2x1 – x2…

A:

Q: Let V be an inner product space, and suppose that T: V→V is linear and that ||T(x)|| = ||x|| for all…

A: Given:

Q: 10. Let X be an inner product space and T: X→→→→→X an isometric linear operator. If dim X<∞, show…

A:

Q: 3.3.9. Given a normed space X, prove that the function p: X R defined by p(x) = ||x|| is continuous…

A:

Q: Y Show that f (2)= 2+ Con formal maps %3D on %3D

A: Given that,f(z)=z+1zz=1

Q: The conjugate z of z=x+iy is z=x-iy

A: 4) Conjugate of a complex number: The complex conjugate is obtained by changing the sign of the…

Q: Theorem 2.9. Suppose p ¢ A in a topological space (X,T). Then p is not a limit point of A if and…

A: Given that, Let (X, T) be the topological space. p is the not element of set A. We have to show…

Q: Prove that if X is a Banach space, X ∕= {0}, then there is a non-zero biounded linear functional on…

A:

Q: Prove that the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.

A: A multiplication operation defined on ℤ/nℤ×ℤ/nℤ→ℤ/nℤ as                x·y=xy

Q: Prove that the family of operators T (t): L2 (0,1) → L2 (0,1), t > 0, where T (t) p (x) = 0 for 0 <r…

A: Let T'=Tt:L20,1→L20,1 be an operator for a fixed t≥0. To prove that this operator is a strongly…

Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C° (R). x In x are linearly independent in…

A: Concept:

Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: 4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T*) and T₁=1 (1 ==…

A:

Q: if T is an operator in a real vectorial space with characteristic polynomial (x²-2x+5)². What are…

A: For the solution follow the next steps.

Q: In the frieze group F7, show that zxz = x-1.

A:

Q: (ii) Verify Cauchy-Goursat theorem for f(z) = 5 sin 2z if C is the square with vertices at 1土も-1土。

A:

Q: Let T: R" → R" be defined as T(x₁,x₂,...,xn) = (x₂ −X₁, X3 X₂, ..., x₁ - xn) of T Discuss the…

A: The matrix is invertible if the determinant of the matrix is non-zero. The matrix is diagonalizable…

Q: Suppose E⊆X , where X is a metric space, p is a limit point of E , f and g are complex functions on…

A:

Q: Prove that if A is invertible, then AT is invertible. What is (AT)-1?

A:

Q: Theorem 4.7. (1) A T2-space (Hausdorff) is a T1-space. (2) A T3-space (regular and T¡) is a…

A: 1)  Consider X is a T2 - space. To prove that, X is a T1- space. By the definition of T2- space,…

Q: Find a norm for C([0, 1]), so that C([0, 1]) is not complete under this norm.

A:

Q: Theorem 7.29. Suppose f : X → Y is a continuous bijection where X is compact and Y is Hausdorff.…

A:

Q: Prove that the family of operators T (t): L2 (0,1) → L2 (0,1), t > 0, where T (t) p (x) = 0 for 0 <r…

A: Fix an operator Tt:L20,1→L20,1, for a fixed t. To prove that Tt is a strongly continuous semigroup.…

Q: use the inner product = p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

A:

Q: Let A ∈ M nxn(F). Show that A is diagonalizable if and only if At isdiagonalizable

A:

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: Determine a scalar s such that A2 x = sx when Ax = r x.

A: Given that Ax=rx. Multiply both sides of the above equation by A and simplify as follows.

Q: . Show that if S is a subbasis for topologies T1 and T2 on a space X, then T1 = T2.

A:

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = 3√x, a = -8

A: Given function is fx=x3 Linearization of function is given as: Lx=fa+f'ax-a At a=-8 f-8=-83=-2…

Q: 4) Suppose that T is an operator on an inner-product space V such that T2 = T. Prove that T is an…

A:

Q: 2. For The Lineor mapping T:RT→R deffined by FLXys,t) = (x-ytstt,' xtas-t, xty+3s-3t) find W OPE a…

A:

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

A:

Q: Use the inner product (f, 9) = f(-1)g(-1) + f(0)g(0) + f(1)g(1) in P, to find the orthogonal…

A: Vbb

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = x + 1/x, a = 1

A: Given fx=x+1x

Q: 1. Show that if & is a function from a nonempty compact metric space X to itself such that d(@(x),…

A:

Q: Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a…

A:

Q: Give an example of a linear operator T on an inner product space V such that N(T) ≠ N(T∗).

A:

Q: Please see image

A: It is given that,

Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

A: ANSWER:

Q: b) Let f(x) = 1 and g(x) = x be functions in C[0, 1]. Use the inner product defined in Example 5, to…

A: Note: Since there is no proper instruction is given and the first part is incomplete, we have solved…

Q: Letf(z) 3z-2i. Find the maximum valne of /(=} on the disk z<.

A: Triangle inequality:z1-z2≤z1+z2 and z1+z2≤z1+z2

Q: Q3- Let S and T be bounded linear operators on a complex Hilbert space H into itself. If (Sx, x) =…

A: Let S and T be bounded linear operator on a complex Hilbert space H into itself. If…

Q: Let X# and t1, t2 are two topologies on X such that 11CT2. Prove or disprove that if (X,t2) is a…

A: Counter Example, The set of real numbers with usual topology. That is,  ℝ,τS usual topology space.…

Q: (b) If f is a bounded linear functional on a Hilbert space H, show that there is a unique y e H such…

A:

Q: Let R2[x]be the vector space of real polynomials of degree < 2. Let f: R2x] - R2x]be the…

A: