Every continuous linear functional ![U[f]](/images/equations/RieszsTheorem/Inline1.svg)
for ![f in C[a,b]](/images/equations/RieszsTheorem/Inline2.svg)
can be expressed as a Stieltjes
integral
![U[f]=int_a^bf(x)dw(x),](/images/equations/RieszsTheorem/NumberedEquation1.svg) |
where 
is determined by 
and is of bounded variation on ![[a,b]](/images/equations/RieszsTheorem/Inline5.svg)
.
Riesz's Theorem
See also
Stieltjes IntegralExplore with Wolfram|Alpha
References
Kestelman, H. "Riesz's Theorem." §11.5 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 265-269, 1960.Referenced on Wolfram|Alpha
Riesz's TheoremCite this as:
Weisstein, Eric W. "Riesz's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RieszsTheorem.html