Prove that l^(infinity), the set of all bounded sequences
equipped with the sup-norm, is not separable. Hint: think
of the decimal expansion proof that the real numbers are
not countable.
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A slight reformulation of the Hahn-Banach Theorem states
that if L is a linear subspace of a normed space X and
f:L -> R is a linear continuous function, then f has a
linear continuous extension F:X -> R with ||F||=||f||,
where ||f|| denotes the norm of f in the dual of L.
(i) Explain why this follows from the Hahn-Banach Theorem
(ii) Prove that this F is unique if H is a Hilbert space.
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Refering to Theorem 1 on
http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01950.JPG,
let X be l^(infinity), the set of all bounded sequences
equipped with the sup-norm, and give an example of an
x for which the linear continuous map F:X -> R is not unique.
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Refering to the proof of the Lemma on
http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01951.JPG,
show that the norm of f in the dual of the space spanned
by L and x_0 is equal to 1.
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Refering to the proof of Thm 2 on
http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01951.JPG,
(i) Prove that if X is a separable normes space, then the
unit sphere {x in X: ||x||=1 } contains a countable set
with closure equal to the unit sphere.
(NB. In the proof (i) is used for the dual of X.)
(ii) Assuming the claim in step 2 to be false,
the lemma implies the existence of an F which is
zero in every x_n and has ||F||=1. Derive a
contradiction.
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Optionary: refering to the proof of Hahn-Banach's Theorem for
separable normed spaces on
http://www.few.vu.nl/~jhulshof/FA2008/week5/DSC01947.JPG,
show that alpha_1 can indeed be chosen as claimed.
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Exercises related to steps in the proof of the closed graph theorem:
Let K be a convex subset of normed space which is symmetric
(-x in K if x is in K). Suppose that K has nonempty interior.
Prove that K contains an open ball centered in the origin.
Let X be a Banach space and x_n in X. If the sum over n of ||x_n||
is finite show that the sum over n of x_n exists in X (i.e.
x_1+...+x_n converges to a limit as n -> infinity). Formulate
and prove the obvious infinite triangle inequality for the sum
of the series.
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Banach-Steinhaus Theorem. Let X and Y be Banach spaces and let S be a
subset of B(X,Y), the Banach space of all linear bounded operators
from X to Y. Suppose that, for every x in X, the set Sx={Ax: A in S}
is bounded. Prove that S is bounded in B(X,Y), i.e. {||A||, A in S}
is bounded. Hint: let W_n={x in X: ||Ax|| < n for all A in S}
Is it necessary for Y to be oomplete?
Apply this theorem and the relation between X and its second dual
to show that weakly convergent sequences are bounded.
Corrollary: prove that weakly convergent sequences are bounded.
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