Riesz representation theorem for Hilbert spaces

Statement

Let $H$ be a real or complex Hilbert space. Adopt the convention that its inner product is linear in the first variable. For every $\varphi$ in the continuous dual $H^*$ , there is a unique $y\in H$ such that

\varphi(x)=\langle x,y\rangle\qquad(x\in H).

Moreover, $\lVert\varphi\rVert=\lVert y\rVert$ . Consequently the map

J:H\longrightarrow H^*,\qquad J(y)(x)=\langle x,y\rangle,

is a conjugate-linear isometric bijection in the complex case and a linear isometric bijection in the real case. This identification depends on the inner product, not only on the normed-space structure.

Proof idea

If $\varphi\neq0$ , its kernel is a closed hyperplane. Choose a nonzero vector $z$ orthogonal to $\ker\varphi$ ; decomposing each $x$ into its kernel component and its component along $z$ produces the representing vector after a scalar normalization. Cauchy--Schwarz gives continuity and the norm identity, while nondegeneracy gives uniqueness Conway, Chapter II, §2.

Consequences

The theorem converts statements about continuous functionals into Hilbert-space geometry. It yields orthogonal projection onto closed subspaces, identifies weak convergence with convergence of all inner products against fixed vectors, and permits the adjoint of a bounded operator to be defined by representing the functional $x\mapsto\langle Tx,y\rangle$ .

Conventions and scope

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990. DOI record. Relevant: Chapter II, §2 on the Riesz representation theorem.
Walter Rudin, Functional Analysis, 2nd ed., McGraw--Hill, 1991. WorldCat record. Relevant: Chapter 4 on Hilbert spaces and continuous linear functionals.