Riesz Representation Theorem

Let $f$ be a bounded linear Functional on a Hilbert Space $H$. Then there exists exactly one $x_0\in H$ such that $f(x)=\left\langle{x, x_0}\right\rangle{}$ for all $x\in H$. Also, $\Vert f\Vert=\Vert x_0\Vert$.

See also Functional, Hilbert Space

References

Debnath, L. and Mikusinski, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990.