The name "Riesz-Fischer theorem" is used in analysis for a number of results concerning the convergence
of Cauchy sequences in Lp
spaces. The theorem is named for mathematicians Frigyes
Riesz and Ernst Fischer, who independently published foundational results for the
special case of 
(Riesz 1907, Fischer 1907).
In its most common 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 in the 
norm to an 
-function 
. This statement is more general than the original results
published by Riesz and Fischer: Riesz's result related convergence
of square-summable sequences of real
numbers to orthonormal sets in ![L^2=L^2([a,b],dx)](/images/equations/Riesz-FischerTheorem/Inline9.svg)
, while Fischer's result proved (using older
terminology and notation) the 
-convergence to a function 
for any Cauchy
sequence in 
.
More general versions are also known. For example, 
is 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. Both of these facts are sometimes considered part of
the Riesz-Fischer theorem.
The name "Riesz-Fischer theorem" is also sometimes used for results which appear 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 spaces and
inner product spaces also attributed to Riesz
and Fischer, a fact sometimes attributable to the connections between the theorem
and its many corollaries.
