Questions tagged [riesz-representation-theorem]

For linear forms in Hilbert spaces: Roughly speaking, the classical Riesz representation theorem says any linear and continuous form on Hilbert space is a scalar product.

For bilinear forms in Hilbert spaces: Let $B:H \times H \to \mathbb{R}$ be a bounded bilinear form on Hilbert space $H$. there eixsts a unique bounded operator $A:H\to H $ such that $B(x,y) = \langle Ax, y\rangle$ for all $x,y\in H.$

For linear forms in $L^p$ spaces: The version of this theorem no $L^p(X,d\mu)$ spaces with $1\le p <\infty$ asserts that:

For every linear and continuous for $\ell\in (L^p(X,d\mu))^*$ there is a unique $g\in L^{p'}(X,d\mu)$ such that $$\ell(f) = \int_X fg d\mu$$ with $\frac{1}{p}+\frac{1}{p'} = 1.$ Patently, this more general than the Hilbert setting $p=2$.

There is another Riesz representation theorem (known as Riesz-Markov theorem) it asserts that:

Riesz-Markov: (for linear forms on Wiener spaces) If $X$ is locally compact Hausdorff space and $\ell : C(X)\to \Bbb R. $ is a linear and continuous form satisfying $\ell(f)\ge 0$ whenever $f\ge 0$. Then there exists a unique Borel measure $\mu$ on $X$ such that $$\ell(f) = \int_X f d\mu, ~~~~\forall~~f\in C(X).$$