Riesz representation theorem

• Preliminaries and notation
  1. Linear and antilinear maps
     • By definition, an antilinear map (also called a conjugate-linear map) $f : \mathcal{H} → Y$ is a map between vector spaces that is additive : $$ f(x + y) = f(x) + f(y) ~~~ \text{for all } x, y ∈ \mathcal{H}, $$ and antilinear (also called conjugate-linear or conjugate-homogeneous) : $$ f(cx) = \bar{c}f(x) ~~~ \text{for all } x ∈ \mathcal{H} \text{ and all scalar } c ∈ \mathbb{F}, $$ where $\mathbb{F}$ is a field, that is, $\mathbb{R}$ or $\mathbb{C}$.
     • In contrast, a map $f : \mathcal{H} → Y$ is linear if it is additive and homogeneous : $$ f(cx) = cf(x) ~~~ \text{for all } x ∈ \mathcal{H} \text{ and all scalar } c ∈ \mathbb{F}.$$ 
  2. Competing definitions of the inner product
     • The map $\langle \cdot, \cdot \rangle$ is linear in its first coordinate. Explicitly, this means that for every fixed $y \in \mathcal{H},$ the map that is denoted by $\left\langle \cdot, y \right\rangle : \mathcal{H} \to \mathbb{F}$ and defined by $$h \mapsto \left\langle h, y \right\rangle ~~~ \text{for all } h \in \mathcal{H}$$ is a linear functional on $\mathcal{H}.$ In fact, this linear functional is continuous, so $\left\langle \cdot, y \right\rangle \in \mathcal{H}^*.$
     • The map $\langle \cdot, \cdot \rangle$ is antilinear in its second coordinate. Explicitly, this means that for every fixed $y \in \mathcal{H},$ the map that is denoted by $\left\langle y, \cdot \right\rangle : \mathcal{H} \to \mathbb{F}$ and defined by $$h \mapsto \left\langle y, h \right\rangle ~~~ \text{for all } h \in \mathcal{H}$$ is a antilinear functional on $\mathcal{H}.$ In fact, this antilinear functional is continuous, so $\left\langle y, \cdot \right\rangle \in \overline{\mathcal{H}}^*.$

 

• Riesz representation theorem
  • Let $\mathcal{H}$ be a Hilbert space whose inner product $\langle x, y \rangle$ is linear in its first argument and antilinear in its second argument. For every continuous linear functional $\varphi ∈ \mathcal{H}^∗,$ there exists a unique vector $f_\varphi ∈ \mathcal{H},$ called the Riesz representation of $\varphi$, such that $$ \varphi(x) = \langle x, f_\varphi \rangle ~~~ \text{for all } x \in \mathcal{H}. $$