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## Subspace Approximation vs. the Riesz-Fischer Theorem

Lecture 8

The previous theorem on page took as given and then constructed a sequence , which by Bessel's inequality

is square-summable, i.e.

By contrast, the Riesz-Fischer theorem on page  allows us to turn this theorem around: Given a square summable sequence,

the R-F theorem considers the concomitant Cauchy sequence in , and then guarantees the existence of a square-intergrable function

This function is related to by virtue of the property that

with the result that

 (115)

Often this is written as

and one says that  equals in the least square sense or in the mean''.

This is to be compared with pointwise equality, which is expressed by the statement that

 (116)

The difference between  '' and  '' manifests itself only when the square-summable sequence yields a function which has one or more discontinuities, then one does not have pointwise equality, Eq.(1.16). Instead, one has the weaker condition, Eq.(1.15). This condition does not specify the value of at the point(s) of discontinuity. Instead, it specifies an equivalence class of functions, all having the same graph everywhere except at the point(s) of discontinuity.

Next: Recapitulation Up: Hilbert Spaces Previous: Approximation via Subspaces: Analysis   Contents   Index
Ulrich Gerlach 2010-12-09