Page 289 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
H : x y,z x,z y,z
1
H : x,y y,x
2
2
H : x,x x
3
for all x,y,z H and for all , C.

Inner Product: Let X be a linear space over the field of complex numbers C. An inner product on
X is a mapping from X × X C which satisfies the following conditions:

(i) ( x + y, z) = (x, z) + (y, z) x, y, z X and , C.

(ii) (x, y) = (y, x)

(iii) (x, x) 0, (x, x) = 0 x = 0
Riesz-representation Theorem for Continuous Linear Functional on a 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 .
*
The Conjugate Space H : 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 the conjugate space of H. The conjugate space of a Hilbert space H is
the conjugate space H of H is in some sense is same as H itself.
*
26.4 Review Questions

1. Let H be a Hilbert space, and show that H* is also a Hilbert space with respect to the inner
product defined by (f , f ) = (y, x). In just the same way, the fact that H* is a Hilbert space
x y
implies that H** is a Hilbert space whose inner product is given by (F , F ) = (g, f).
f g
2. Let H be a Hilbert space. We have two natural mappings of H onto H**, the second of
which is onto: the Banach space natural imbedding x F , where f (y) = (y, x) and
x x
F (f) (F, f ). Show that these mappings are equal, and conclude that H is reflexive. Show
f x x
that (F , F ) = (x, y).
x y
26.5 Further Readings

Books Hausmann, Holm and Puppe, Algebraic and Geometric Topology, Vol. 5, (2005)
K. Yosida, Functional Analysis, Academic Press, 1965.

Online links www.spot.colorado.edu

www.arvix.org

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