Page 231 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Also x a x i is bounded … (1)
1 2
i is continuous.
Hence by open-mapping theorem, i is open and so it is homeomorphism of (N, x ) onto
2
(N, x ). Consequently i is bounded as a map from (N, x ) (N, x )
1 1 2
–1
Since i (x) = x, a, b s.t. x b … (2)
2 1
(1) & (2) imply that the norms are equivalent.
19.2 Summary

 If B and B are Banach spaces and T is a continuous linear transformation of B onto B , then
the image of each sphere centered on the origin in B contains an open sphere centered on
the origin in B .
 The open mapping theorem : If T is a continuous linear transformation of a Banach space
B onto a Banach space B , then T is an open mapping.

19.3 Keywords

Banach Space: A normed space V is said to be Banach space if for every Cauchy sequence
V then there exists an element V such that lim .
n n 1 n n
Homeomorphism: A map f : (X, T) (Y, U) is said to be homeomorphism if

(i) f is one-one onto.
–1
(ii) f and f are continuous.
+
Open Sphere: Let x X and r R . Then set {x X : p (x , x) < r} is defined as open sphere with
o o
centre x and radius r.
o
19.4 Review Questions

1. If X and Y are Banach spaces and A : X Y is a bounded linear transformation that is
bijective, then prove that A is bounded.
–1
2. Let X be a vector space and suppose , and are two norms on X and that T and T
1 2 1 2
are the corresponding topologies. Show that if X is complete in both norms and T T ,
1 2
then T = T .
1 2
19.5 Further Readings

Books Walter Rudin, Functional Analysis, McGraw-Hill, 1973.
Jean Diendonne, Treatise on Analysis, Volume II, Academic Press (1970).

Online links euclid.colorado.edu/ngwilkin/files/math 6320…/OMT_CGT.pdf
people.sissa.it/nbianchin/courses/…/lecture05.banachstein.pdf

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