Page 355 - Advanced Linear Algebra
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Hilbert Spaces 339
or equivalently,
B
( (º%Á " » % )
)
~
with equality if and only if % cspan ²E . ³
The Arbitrary Case
To discuss the case of an arbitrary orthonormal set E ~¸" 2¹ , let us
first define and discuss the concept of the sum of an arbitrary number of terms.
(This is a bit of a digression, since we could proceed without all of the coming
details c but they are interesting. ³
Definition Let A ~¸% 2¹ be an arbitrary family of vectors in an inner
product space . The sum
=
%
2
is said to converge to a vector %= and we write
%~ % (13.7 )
2
if for any , there exists a finite set : 2 for which
; :Á ; finite ¬ % c %
i
i
;
For those readers familiar with the language of convergence of nets, the set
(
F ²2³ of all finite subsets of 2 is a directed set under inclusion for every
(Á ) F ²2³ there is a * F ²2³ containing ( and ) and the function)
:¦ %
:
is a net in . Convergence of 13.7 is convergence of this net. In any case, we
)
/
²
will refer to the preceding definition as the net definition of convergence.
It is not hard to verify the following basic properties of net convergence for
arbitrary sums.
Theorem 13.19 Let A ~¸% 2¹ be an arbitrary family of vectors in an
inner product space . If
=
%~ % and & ~ &
2 2
then
