Page 234 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 20: The Closed Graph Theorem

Notes

Notes The following norms are equivalent to above norm
(i) (x, y) = max { x , y }
(ii) (x, y) = x + y (p = 1 in the above theorem)

20.1.2 Closed Linear Transformation

Definition: 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
Theorem 2: Let N and N be normed linear spaces and B be a subspace of N. Then a linear
transformation T : M N is closed its graph G is closed.
T
Proof: Let T is closed linear transformation. We claim that its graph G is closed i.e. G contains
T T
all its limit point.
Let (x, y) be any limit point of G . Then a sequence of points in G , (x , T (x ), x M, converging
T T n n n
to (x, y). But
(x , T (x )) (x, y)
n n
x , T (x ) – (x, y) 0
n n
(x – x), T (x ) – y 0
n n
x – x + T (x ) – y 0
n n
x – x 0 and T (x ) – y 0
n n
x x and T (x ) y ( T is closed)
n n
(x, y) G . (By def. of graph)
T
Thus we have shown that every limit point of G is in G and hence G is closed.
T T T
Conversely, let the graph of T, G is closed.
T
To show that T is closed linear transformation.
Let x M, x x and T (x ) y.
n n n
Then it can be seen that (x, y) is an adherent point of G so that
T
(x, y) G . But G = G ( G is closed)
T
T
T T
Hence (x, y) G and so by the definition of G we have x M and y = T (x).
T T
Consequently, T is a closed linear transformation. This completes the proof of the theorem.
20.1.3 The Closed Graph Theorem – Proof

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
Proof: Necessary Part:
Let T be continuous and let G denote the graph of T, i.e.
T
G = {(x, T (x) : x B} B × B .
T

LOVELY PROFESSIONAL UNIVERSITY 227