Riesz Representation Theorem

Every Linear Functional on ${\mathbb{C}}^n$ or ${\mathbb{R}}^n$ is of the form of a Standard Inner Product Function.

Theorem

• If $f\in \mathcal{L}({\mathbb{C}}^n,\mathbb{C})$
• Then, $\exists \alpha$ s.t $f(v)=f_{\alpha }(v),\forall v\in V$
  • Note that $f_{\alpha }$ is a Standard Inner Product Function

Proof

• Let $c\in \mathbb{C}$ be arbitrary
• Let $x,y\in {\mathbb{C}}^n$ be arbitrary
• Then, $x=(x_1,\dots ,x_n)$
• Then, $y=(y_1,\dots ,y_n)$
• Note $f_{\alpha }(cx+y)=⟨(cx+y)|\alpha ⟩$
  • $={\sum }_{i=1}^n(cx+y{)}_i{\overline{\alpha }}_i$
  • $={\sum }_{i=1}^{n}c{x}_i{\overline{\alpha }}_i+y_i{\overline{\alpha }}_i$
  • $=c{\sum }_{i=1}^n{x}_i{\alpha }_i+{\sum }_{i=1}^n{y}_i{\alpha }_i$
  • $=c⟨x|a⟩+⟨y|a⟩$
  • $=c{f}_{\alpha }(x)+f_{\alpha }(x)$