Page 348 - Advanced Linear Algebra
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332 Advanced Linear Algebra
Figure 13.1
,
Let us see what we can learn about this sequence. First, if we let &~ % c
then according to the parallelogram law,
) )&b b )& )&c & ) ) ~ ² )& ) b & ³
or
) )&c & ) ) ~ ² )& ) b & ³ c h &b & h (13.2 )
Now, if the set is convex , that is, if
:
%Á & : ¬ % b ² c ³& : for all
(in words, contains the line segment between any two of its points ), then
:
² b ³° : and so
&b & b
h h ~% h c
h
Thus, 13.2 gives
(
)
) )&c & ) ) ² )& ) b & ³ c ¦
as Á ¦ B . Hence, if is convex, then the sequence ²& ³ ~ ²% c ³ is a
:
Cauchy sequence and therefore so is ² ³ .
If we also require that be complete, then the Cauchy sequence ² ³ converges
:
)
V
to a vector %: and by the continuity of the norm, we must have % c % ~ .
V
)
Let us summarize and add a remark about uniqueness.
Theorem 13.9 Let be an inner product space and let be a complete convex
:
=
=
subset of . Then for any % = , there exists a unique V : for which
%
) V )%c% ~ inf ) )%c
:
The vector is called the best approximation to in .
%
:
% V
