In analysis, the phrase "Riesz-Fischer theorem" is used to describe a number of results concerning the convergence of Cauchy
sequences in L-p spaces. The theorem is named
for mathematicians Frigyes Riesz and Ernst Fischer who in 1907 independently published
foundational results for the special case of 
.
In its most often-utilized form, the Riesz-Fischer theorem says that for 
, the space 
is (sequentially) complete
for all measure spaces 
, i.e., that every Cauchy
sequence of functions in 
converges to an 
-function 
. This statement is far more general than the original results
published by Riesz and Fischer, however, as Riesz's result related convergence of
square-summable sequences of real numbers to orthonormal systems in ![L^2=L^2([a,b],dx)](/images/equations/Riesz-FischerTheorem/Inline8.svg)
while Fischer's result proved (using more
antiquated terminology and notation) the 
-convergence to a function 
for any Cauchy sequence in 
.
It is worth noting that even more generality may be added to the above statements. For example, 
is also known to be complete for 
. Moreover, given an inner
product space 
of functions and an orthonormal
set 
in 
,
the above-stated theorem can be generalized to show that sequential completeness
of 
(i.e., 
being a Hilbert space) implies the existence of
a limiting function 
for every square-summable sequence with finite 
norm. Though done somewhat less commonly, both of these
facts may be considered part of the Riesz-Fischer theorem.
It is also worth noting that the phrase "Riesz-Fischer theorem" is sometimes used for results which appear altogether dissimilar to the above. Because of the
role played by 
in the study of Fourier series, for example, it
is not uncommon to see literature attribute to Riesz and Fischer various Fourier-theoretic
results regarding convergence of square integrable
functions. It is also somewhat common - particularly in older literature - to see
a number of seemingly-unrelated facts about normed linear and inner product spaces also attributed to Riesz
and Fischer, a fact sometimes attributable to the subtle connections between the
actual theorem and its many corollaries.
