Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.
i) X is a Banach space.
ii) Y is a Banach space.
iii) F is a closed map.
iv) Which property of continuity is being established to conclude that F is continuous.
In the proof of the closed graph theorem, we consider a linear operator (F: X \to Y) between Banach spaces (X) and (Y), and assume that the graph of (F), (G(F) = {(x, F(x)) : x \in X}), is closed in (X \times Y). The goal is to show that (F) is continuous. i) Use of (X) being a Banach space: We define a sequence ((x_n)) in (X) such that (x_n \to x) in (X) and (F(x_n) \to y) in (Y). To conclude that (x) actually belongs to (X) (so the limit exists in (X)) and to apply _______ _________ ______ ________ ___ _______ ________ __________.
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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).
Let X be a Banach space and Y be a closed subspace of X. Let be canonical quotient map. Show that
is open.
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
) 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.
Prove the following result:
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
. 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.
Let X be a normed space and Y be proper subspace of X. Show that the interior Y0 of Y is empty.