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Measure Theory and Functional Analysis
Notes Reflexive Mapping: If the map J : N N** defined by
J (x) = F x N,
x
is onto also, then N (or J) is said to be reflexive (or reflexive mapping).
18.4 Review Questions
1. Let X be a compact Hausdorff space, and justify the assertion that C (X) is reflexive if X is
finite.
2. If N is a finite-dimensional normed linear space of dimension n, show that N* also has
dimension n. Use this to prove that N is reflexive.
3. If B is a Banach space, prove that B is reflexive B* is reflexive.
4. Prove that if B is a reflexive Banach space, then its closed unit sphere S is weakly compact.
5. Show that a linear subspace of a normed linear space is closed it is weakly closed.
18.5 Further Readings
Books G.F. Simmons, Introduction to topology and Modern Analysis. McGraw-Hill,
Kogakusha Ltd.
J.B. Conway, A Course in Functional Analysis. Springer-Verlag.
Online links www.mathoverflow.net/…/natural-embedding
www.tandfonline.com
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