Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the...

Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the norm of projection Pe(x) =1 except for trivial case when e = {0}.

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Colby Messinger answered 2 years ago

7. Show that the dual space H' of a Hilbert space H is a Hilbert space...

7. Show that the dual space H' of a Hilbert space H is a Hilbert space with inner product (', ')1 defined by (f .. fV)1 = (z, v)= (v, z), where f.(x) = (x, z), etc.

Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed,...

Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed, so is R(A*)

2. (a). Prove that the sum of the two projection operators is not a projection operator,...

2. (a). Prove that the sum of the two projection operators is not a projection operator, unless the multiplication of the two projection operators produces a zero value. (b). Prove that the result of the multiplication of two projection operators is not a projection operator, unless the two projection operators are hermitian.

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that...

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that T is diagonalizable if and only if for every λ ∈ C, we have N(T − λIV ) = N((T − λIV )2).

hzhshs assume that X is normed linear space let A is a subset of X prove...

hzhshs assume that X is normed linear space let A is a subset of X prove that: if A is compact then A is closed and bounded

Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the...

Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the definition of Frechet differentiation, show that ∇f(x) = x for all x ̸= 0. Furthermore, show that f(x) is not Frechet differentiable at x = 0.

(a)For what range in ν is the function f(x) = x ν in Hilbert space? (4)...

(a)For what range in ν is the function f(x) = x ν in Hilbert space? (4) (b) Why are observables represented by Hermitian operators? Explain fully. (3) (c) Why are determinate states of Q eigenfunctions of Qˆ? Explain fully. (4) (d) Comment on the essential properties of reality, orthogonality and completeness for both the cases of discrete and continuous spectra.

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Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the...

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