Riesz representation theorem

For a real or complex Hilbert space $H$ with inner product linear in its first argument, every bounded linear functional $f:H\to \mathbb{F}$ has a unique representing vector $y\in H$ such thatThe Cauchy-Schwarz inequality gives $\Vert f\Vert \le \Vert y\Vert$, and evaluation at $x=y/\Vert y\Vert$ gives equality when $y\ne 0$. Uniqueness follows because $⟨x,y-z⟩=0$ for every $x$ implies $y=z$. Existence is proved by the Kernel-orthogonal proof of the Riesz representation theorem.^[ref]

For example, on $L^2(0,1)$, $f(g)={\int }_0^{1}tg(t)dt$ is represented by $y(t)=t$, with $\Vert f\Vert =1/\sqrt{3}$. By contrast, point evaluation $g\mapsto g(0)$ is not a bounded functional on this L2 space: elements are equivalence classes modulo equality almost everywhere, so point values are not even well-defined. On continuous functions, triangular peaks can also have value one at zero and arbitrarily small L2 norm.

On a complex Hilbert space, $y\mapsto ⟨\cdot ,y⟩$ is conjugate-linear. Thus the representation is an isometric conjugate-linear identification with the continuous dual space, under the declared convention; it should not silently be called a complex-linear map.

The Riesz representation theorem is a fundamental result in functional analysis that characterizes certain types of linear functionals on a space of continuous functions. The most commonly referenced version of the theorem deals with the space of continuous functions on a compact Hausdorff space, often denoted as \( C(X) \), where \( X \) is a compact Hausdorff topological space.