Page 243 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes This completes the proof of the theorem.
Theorem 4: Let T be an operator on a Banach space B. Then T has an inverse T–1 T* has an
inverse (T*) , and
–1
(T*)–1 = (T–1)*
–1
–1
–1
Proof: T has inverse T TT = T T = I
By theorem 2, the mapping : T T* reverse the product and preserves the identity
(TT )* = (T T)* = I*
–1
–1
(T )*T = T* (T )* = I
–1
–1
(T*) exists and it is given by (T*) = (T )*. This completes the proof of the theorem.
–1
–1
–1
21.2 Summary
+
 Let N be the linear space of all scalar-valued linear functions defined on N. Clearly the
+
conjugate space N* is a subspace of N . Let T be a linear transformation T of N into itself
as follows:
+
If f N , then T (f) is defined as
T (f) x = f (T (x))
 Let N be a normed linear space and let T be a continuous linear transformation of N into
itself. Define a linear transformation T* of N* into itself as follows:

+
If f N , then T (f) is given by
T (f) x = f (T (x))
We call T* the conjugate of T.

21.3 Keywords

The Conjugate of T: Let N be normed linear space and let T be a continuous linear transformation
of N into itself (i.e. T is an operator). Define a linear transformation T* of N* into itself as
follows:

If f N*, then, T* (f) is given by
[T* (f)] (x) = f (T (x))
We call T* the conjugate of T.

The Linear Function: Let N* be the linear space of all scalar-valued linear functions defined on N.
Clearly the conjugate space N* is a subspace of N*. Let T be a linear transformation T of N* into
itself as follows:

+
If f N , then T (f) is defined as
[T (f)]x = f (T (x))
+
Since f (T (x)) is well defined, T is a well-defined transformation on N .
21.4 Review Questions

1. Let B be a Banach space and N a normed linear space. If {T } is a sequence in B (B, N) such
n
that T(x) = lim T (x) exists for each x in B, prove that T is a continuous transformation.
n

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