Consider the space , define
. Show that f is a linear functional which is not continuous w.r.t the norm
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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.
Give an example of a compact linear map on l2.
Consider the space , define
. Show that f is a linear functional which is not continuous w.r.t the norm
Given an example of an Hilbert space H and an operator A on Η such that σe(A)is empty. Justify your choice of example.
Let X be a Banach space, Y be a normed linear space and be a subset of B(X, Y). If
is not uniformly bounded, then there exists a dense subset D of X such that for every
is not bounded in Y.
Which of the following maps are open? Give reasons for your answer.
i)
ii)
Let X,Y be normed spaces and suppose BL(X,Y) and CL(X,Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X,Y) is linear subspace of BL(X,Y). Also, Show that if Y is a Banach space, then CL(X,Y) is a closed subspace of BL(X,Y).