Chapter 4 Hilbert Space

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Chapter 4 Hilbert Space
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4.1 Inner product space
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Inner product
E complex vector space
is called an inner product on E if
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Inner product space
E complex vector space
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is an inner product on E
With such inner product E is called
inner product space. If we write
,then
is a norm on E and hence
E is a normed vector space.
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Schwarz Inequality
E is an inner product space
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Triangular Inequality for ? .?
E is an inner product space
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Example 1 for Inner product space
Let
For
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Example 2 for Inner product space
Let
For
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Example 3 for Inner product space
Let
For
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Exercise 1.1 (i)
For
Show that
and hence
is absolutely convergent
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Exercise 1.1 (ii)
Show that
is complete
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Hilbert space
An inner product space E is called
Hilbert space if
is complete
is a Hilbert space of which
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Exercise 1.2
Define real inner product space and
real Hilbert space.
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4.2 Geometry for Hilbert space
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Theorem 2.1 p.1
E inner product space
M complete convex subset of E
Let
then the following are equivalent
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Theorem 2.1 p.2
(1)
(2)
Furthermore there is a unique
satisfing (1) and (2).
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Projection from E onto M
The map
defined by txy, where
y is the unique element in M which satisfies (1)
of Thm 1 is called the projection from E onto M.
and is denoted by
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Corollary 2.1
Let M be a closed convex subset of a Hilbert
space E, then
has the following
properties
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Convex Cone
A convex set M in a vector space is called
a convex cone if
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Exercise 2.2 (i)
Let M be a closed convex cone in a Hilbert
space E and let
Put
Show that
I being the identity map of E.
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Exercise 2.2 (ii)
( t is positive homogeneous)
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Exercise 2.2 (iii)
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Exercise 2.2 (iv)
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Exercise 2.2 (v)
conversely if
then
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Exercise 2.2 (vi)
In the remaining exercise, suppose that
M is a closed vector subspace of E. Show that
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Exercise 2.2 (vii)
both t and s are continuous and linear
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Exercise 2.2 (viii)
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Exercise 2.2 (ix)
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Exercise 2.2 (x)
tx and sx are the unique elements
such that xyz
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4.3 Linear transformation
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We consider a linear transformation from
a normed vector space X into a normed
vector space Y over the same field R or C.
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Exercise 1.1
T is continuous on X if and only if
T is continuous at one point.
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Theorem 3.1
T is continuous if and only if there is a
such that
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Theorem 3.3Riesz Representation Theorem
Let X be a Hilbert space and
such that
then there is
Furthermore the map
is conjugate linear and
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4.4 Lebesgue Nikondym Theorem
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Indefinite integral of f
be a measurable space
Let
and f a S measurable function on O
Suppose that
has a meaning
then the set function defined by
is called
the indefinite integral of f.
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Property of Indefinite integral of f
? is s- additive i.e. if
is a disjoint sequence, then
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Absolute Continuous
? is said to be absolute continuous w.r.t µ if
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Theorem 4.1Lebesgue Nikodym Theorem
with
Suppose that ?is absolute continuous w.r.t. µ,
then there is a unque
such that
Furthermore
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4.5 Lax-Milgram Theorem
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Sesquilinear p.1
Let X be a complex Hilbert space.
is called sesquilinear if
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Sesquilinear p.2
B is called bounded if there is rgt0 such that
B is called positive definite if there is ?gt0 s.t.
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Theorem 5.1The Lax-Milgram Theorem p.1
Let X be a complex Hilbert space and B a
a bounded, positive definite sesquilinear
functional on X x X , then there is a unique
bounded linear operator SX ?X such that
and
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Theorem 5.1 The Lax-Milgram Theorem p.2
Furthermore
exists and is bounded with
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4.7 Bessel Inequality and parseval Relation
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Propositions p.1
be an orthogonal system in a
Let
Hilbert space X, and let U be the closed vector
subspace generated by
Let
be the orthogonal projection onto U
where
and
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Proposition (1)
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Proposition (2)
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Proposition (3)
For each k and x,y in X
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Proposition (4)
For any x,y in X
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Proposition (5)
Bessel inequality
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Proposition (6)
( Parseval relation)
An orthonormal system
is called complete if UX
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Separable
A Hilbert space is called separable
if it contains a countable dense subset
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Theorem 7.1
A saparable Hilbert space is isometrically
isomorphic either to
for some n
or to
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