Page 237 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes  Let N and N be normed linear spaces and let M be a subspace of N. Then a linear
transformation T : M N is said to be closed iff x M, x x and T (x ) y imply
n n n
x M and y = T (x).
 If B and B are Banach spaces and if T is linear transformation of B into B , then T is
continuous Graph of T (G ) is closed.
T
20.3 Keyword

Closed Linear Transformation: Let N and N be normed linear spaces and let M be a subspace of
N. Then a linear transformation
T : M N is said to be closed

iff x M, x x and T (x ) y imply x M and y = T (x).
n n n
20.4 Review Questions

1. If X and Y are normed spaces and A : X Y is a linear transformation, then prove that
graph of A is closed if and only if whenever x 0 and Ax y, it must be that y = 0.
n n
2. If P is a projection on a Banach space B, and if M and N are its range and null space, then
prove that M and N are closed linear subspaces of B such that B = M N.

20.5 Further Readings

Books Folland, Gerald B, Real Analysis: Modern Techniques and their Applications (1st ed.),
John Wiley & Sons, (1984).
Rudin, Walter, Functional Analysis, Tata McGraw-Hill (1973).

Online links euclid.colorado.edu/ngwilkin/files/math6320.../OMT_CGT.pdf
mathworld.wolfram.com

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