Page 309 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes We have I T x 0 Ix Tx 0 x Tx 0
Tx x

Tx,x x,x x 2
2
x 0  Tx,x 0
2
x 0
2
x 2 0  x is always 0
x 0
I T x 0 x 0.

Now I T x I T y I T x y 0

x y 0 x y
Hence I+T is one-one.
I+T is onto.

Let M = range of I+T. Then I+T will be onto if we prove that M=H.
We first show that M is closed.
For any x H, we have

2 2
I T x x Tx
x Tx,x Tx
x,x x,Tx Tx,x Tx,Tx

x 2 Tx 2 Tx,x Tx,x

x 2 Tx 2 2 Tx,x  T is positive T is self-adjoint Tx,x real

x 2  T is positive Tx,x 0

Thus x I T x x H
Now let I T x be a CAUCHY sequence in M. For any two positive integers m,n we have
n

x x I T x x
m n m n
I T x m I T x n 0,

since I T x is a CAUCHY sequence.

x x 0
m n
x is a CAUCHY sequence in H. But H is complete. Therefore by CAUCHY sequence x in
n n
H converges to a vector, say x in H.

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