Give an example of a compact linear map on l2.
Consider the sequence space l2l^2, the space of square-summable sequences of real or complex numbers. Let T:l2→l2T: l^2 \to l^2 be a linear map defined by:
T((xn))=(x12,x24,x38,… )T((x_n)) = \left( \frac{x_1}{2}, \frac{x_2}{4}, ________ _________ _________ _______ ________ __________ ___ _________ _________ ________ ______.
__________ __________ ___ _________ _____ ______ ______ __________.
______ ________ ____ _________ _____ _______ ___ __________ ________ _________ __________ _______.
_______ _____ ___ _____ _______ ___ ___ _____ _____.
________ ____ ___ _____ _________ __________ _____ ______ ________ ___.
______ _____ _________ ______ ___ _______.
_____ ____ ________ ____ ______ ___ _____ ________ _____.
______ ________ ___ ______ ________ ______ ___.
_______ _________ __________ _________ _______ _________ ___ _______.
__________ _________ ________ _______ ________ __________ __________ __________ _______.
_________ ________ __________ ___ ___ ____.
_______.
Get Full Answer on WhatsApp
Give one example of each of the following. Also justify your choice of example.
i) A self-adjoint operator on .
ii) A normal operator on a Hilbert space which is not unitary.
Let X = C00 with Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.
Prove the following result:
Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set {r1,r2,..., rn} of a non-zero real numbers with and an orthonormal set {w1, w2,..., wn} in H such that
Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?
Consider the space C1[0,1] of all C1 functions on [0,1] endowed with the uniform norm induced from the space C[0,1], and consider the differential operator defined by Df = f'. Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C1[0, 1] is not a Banach space? Justify your answer.
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
b) Co is a Banach space.
c) If A is the right shift operator on l2, then the eigen spectrum is non-empty.
d) If a normed linear space is reflexive, then so is its dual space.
e) If a normed linear space X is finite dimensional, then so is X'.
Let f: C[0,1]→ be given by f (x) = x(1)∀x ∈ C[0,1]. Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.
Let A be a normal operator on a Hilbert space X. Show that σ(A) ⊂ σa(A) where σa (A) denotes the approximate eigen spectrum of A and σ(A) denotes the spectrum of A.