Page 350 - Advanced Linear Algebra
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334 Advanced Linear Algebra
in which case
(º%Á »
(
) )%c ) ) ~ % c
))
%
Replacing by c % V gives
%
( V (º% c %Á »
) )%c%c ) V ) ~ % c% V c
))
%
But is the best approximation to in and since c % V : % V : we must have
) )%c%c ) V ) % c% V
Hence,
( V (º% c %Á »
~
))
or equivalently,
º% c %Á » ~
V
V
Hence, %c% : .
According to Theorem 13.9, if is a complete subspace of an inner product
:
space , then for any % = , we may write
=
V
%~% b ²% c %³ V
where % : and % c % : . Hence, = ~ : b : and since : q : ~ ¸ ¹ ,
V
V
we also have =~ : p : . This is the projection theorem for arbitrary inner
product spaces.
Theorem 13.11 The projection theorem) If is a complete subspace of an
(
:
inner product space , then
=
=~ : p :
In particular, if is a closed subspace of a Hilbert space , then
:
/
/~ : p :
Theorem 13.12 Let , and be subspaces of an inner product space .
Z
=
:;
;
)
1 If =~ : p ; then ; ~ : .
)
2 If :p ; ~ :p ; Z then ; ~ ; Z .
Proof. If =~ : p ; , then ; : by definition of orthogonal direct sum. On
the other hand, if ' : , then '~ b ! , for some : and ! ; . Hence,
~ º'Á » ~ º Á » b º!Á » ~ º Á »
