Page 288 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 26: The Conjugate Space H*

By definition of J, J(x) = F . Hence to show T .T = J, we have to prove that F = F . Notes
x 2 1 x f x
*
For this let f H . Then f = f where f corresponds to y in the representation
y
F (f) (f,f ) (f ,f ) (x,y) .
f x x y x
But (x,y) = f (x) = f(x) = F (f).
y x
*
Thus we get F (f) F (f) for every f H .
f x x
Hence the mapping F and F are equal.
f x x
T .T = J and J is a mapping of H onto H , so that H is reflexive.
**
2 1
This completes the proof of the theorem.

Notes

1. Since F F x H (From above theorem)
x f x
F ,F y F ,F f y f ,f x x,y by using def. of inner product on H and by the
**
x
y
f y
def. of inner product on H . *
**
2. Since an isometric isomorphism of the Hilbert space H onto Hilbert space H ,
**
therefore we can say that Hilbert space H and H are congruent i.e. they are equivalent
**
metrically as well as algebraically. We can identify the space H with the space H.
26.2 Summary
 Let H be a Hilbert space. If f is a functional on H, then f will be continuous linear functional
on H. The set H,C of all continuous linear functional on H is denoted by H and is called
*
*
conjugate space of H. Conjugate space of a Hilbert space H is the conjugate space H of H.
 Riesz-representation theorem for continuous linear functional on Hilbert space:
*
Let H be a Hilbert space and let f be an arbitrary functional on H . Then there exists a
unique vector y in H such that f = fy, i.e. f(x) = (x,y) for every vector x H and f y .

26.3 Keywords

Continuous Linear Functionals: Let N be a normal linear space. Then we know that the set R of
real numbers and the set C of complex numbers are Banach spaces with the norm of any
x R or x C given by the absolute value of x. We denote the BANACH space N,R or N,C
by N . *

The elements of N will be referred to as continuous linear functionals on N.
*
Hilbert space: A complete inner product space is called a Hilbert space.
Let H be a complex Banach space whose norm arises from an inner product which is a complex
function denoted by (x,y) satisfying the following conditions:

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