Give an example of a compact linear map on l2.
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State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
b) C0 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 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.
Let X be a Banach space and Y be a closed subspace of X. Let be canonical quotient map. Show that
is open.
Consider the space c00. For , define
. Show that f is a linear functional which is not continuous w.r.t the norm
.
Let X be an inner product space and . Prove that
if and only if
.
) Which of the following maps are open? Give reasons for your answer.
i) given by
.
ii) given by
.
Give an example of a compact linear map on l2.