Page 356 - Advanced Linear Algebra
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340 Advanced Linear Algebra
1)(Linearity )
² % b & ³ ~ % b &
2
for any Á -
2)(Continuity )
º% Á&» ~ º%Á&» and º&Á% » ~ º&Á%»
2 2
The next result gives a useful “Cauchy-type” description of convergence.
Theorem 13.20 Let A ~¸% 2¹ be an arbitrary family of vectors in an
inner product space .
=
)
1 If the sum
%
2
converges, then for any , there exists a finite set 0 2 such that
1q 0 ~ JÁ 1 finite ¬ % i
i
1
)
)
2 If is a Hilbert space, then the converse of 1 also holds.
=
)
:
Proof. For part 1 , given , let : 2 , finite, be such that
; :Á ; finite ¬ % c %
i
i
; 2
If 1q : ~ J , finite, then
1
i i i % ~ b ² % % ³ c ² % % c % ³ i c
1 1 : :
i i i % c% b % c% 2 b 2 ~
i
1r: :
)
As for part 2 , for each , let 0 2 be a finite set for which
1q 0 ~ JÁ 1 finite ¬ % i
i
1
and let
&~ %
0
