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Every continuous linear functional for can be expressed as a Stieltjes integral
where is determined by and is of bounded variation on .
![U[f]](/images/maths/functional-analysis/rieszs/Inline1.gif)
for
![f in C[a,b]](/images/maths/functional-analysis/rieszs/Inline2.gif)
can be expressed as a
Stieltjes integral 
is determined by
and is of bounded variation on
![[a,b]](/images/maths/functional-analysis/rieszs/Inline5.gif)
.
![U[f]=int_a^bf(x)dw(x),](/images/maths/functional-analysis/rieszs/NumberedEquation1.gif)
