Let T: H- » H be a bounded positive self-adjoint linear operator on a complex Hilbert space. Using the positive square root of T, show that for all x, y e H, (Tx, y)|S(Tx, x)/2(Ty, y)'².

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Q: Let f,g: [a, b]-R are Riemann integrable f-g is Riemann integrable.

A: Using properties of Riemann Sum we can say f-g will be integrable.

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A: Spectral theorem for Hermitian operators H in a finite dimensional inner product space V:…

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

A: True

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Q: Q6: Show that R³ = {(x1,x2, x3)|lx1,x2, x3 E R} is a vector space

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Q: Let U be the vector space of 3×2 matrices over R. Show that dim.U =6.

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Q: ITA Is a bounded self-adjoint operator on a Hilbert space н, show that |= sup {|}: xe H |x| = 1}

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Q: 1. If two Hermitian operators A and B commute, show that they can be simultaneously diagonalized.

A: Given :-  If two Hermitian operators A and B commute , then they can be simultaneously diagonalized.

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A: Trick here is to show that H has an eigenvalue and if H has an eigenvalue then by the definition of…

Q: Let V be a vector space over R, and dimV=200. Let NEL(V) be a nilpotent operator such that null(N2)…

A: Consider the given information.

Q: Let V be a finite dimensional vector space, and T:        V------->V be a linear operator. Let NT…

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Q: Suppose that you have three vectors: f1 (x) = 1, f2 (x) = x – 1, and f3 (x) = ; (x² – 4x + 2), that…

A: Suppose that you have three vectors: f1(x)=1, f2(x)=x-1, and f3(x)=12x2-4x+2, that make up an…

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A: T is an operator in an inner product space V over the field F together with an inner product. V×V→F…

Q: give an example of an infinite dimensional vector space V over complex numbers and a linear operator…

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Q: Theorem 3. Let A be a bounded linear operator acting between two real Hilbert spaces. Then 1.…

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

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Q: Find an orthonormal basis of the plane x1 - 6x2 - x3 = 0. { }

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Q: Let A be an m x n matrix with orthonormal columns, and x, y be in R" then prove tat the linear…

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Q: Let V be a finite dimensional vector space, and T:        V------->V be a linear operator. Let NT…

A: Given: V be a finite-dimensional vector space, and T: V--->V be a linear operator. Let NT denote…

Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if omplex…

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Q: Consider inner-product = | f(x)g(x) dx in vector space C[- n, T], the Fourier expansion of f(x) = x…

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Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if eomplex…

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Q: Give example on the inner product and same time is normed space

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Q: Use Laplace transform to find a fundamental set of vectors for the system of ODES %3D

A: Step:-1 Note:- Lf'(t)=s F(s)-F(0) Given that x→'=1221x→ Let  x→(t)=x(t)y(t), Here we take t as…

Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if --- omplex…

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Q: A real-valued function f defined on the real line is called an even function if  J( -t) = f (t) for…

A: Givenf(-t)=f(t)-even functions.To prove Set of even functions forms a vector space.

Q: Find the standard matrix for the stated composition of linear operators on, orthogonal projection on…

A: Reflection Matrix about the xy-plane is : Rxy=10001000-1 Reflection Matrix about the xz-plane is :…

Q: now that Reis Banach f|xl|= | ( 2 | ², 1² ) / 2 j=1 and vector space. 2h is complete.

A: We have given Rn andx=∑j=1n  ξiq12. We have to show that Rn is Banch space, vector space and…

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…