Mathematical Methods of Physics/Riesz representation theorem

In this chapter, we will more formally discuss the bra $$|\rangle$$ and ket $$\langle |$$ notation introduced in the previous chapter.

Definition
Let $$\mathcal{H}$$ be a Hilbert space and let $$\ell:\mathcal{H}\to\mathbb{C}$$ be a continuous linear transformation. Then $$\ell$$ is said to be a linear functional on $$\mathcal{H}$$.

The space of all linear functionals on $$\mathcal{H}$$ is denoted as $$\mathcal{H}^*$$. Notice that $$\mathcal{H}^*$$ is a normed vector space on $$\mathbb{C}$$ with $$\|\ell\|=\sup\left\{\frac{|\ell(x)|}{\|x\|}:x\in\mathcal{H}; \|x\|\neq 0\right\}$$

We also have the obvious definition, $$\mathbf{a},\mathbf{b}\in\mathcal{H}$$ are said to be orthogonal if $$\mathbf{a}\cdot\mathbf{b}=0$$. We write this as $$\mathbf{a}\perp\mathbf{b}$$. If $$A\subset\mathcal{H}$$ then we write $$\mathbf{b}\perp A$$ if $$\mathbf{b}\perp\mathbf{a}\forall\mathbf{a}\in A$$

Theorem
Let $$\mathcal{H}$$ be a Hilbert space, let $$\mathcal{M}$$ be a closed subspace of $$\mathcal{H}$$ and let $$\mathcal{M}^{\perp}=\{x\in\mathcal{H}:(x\cdot a)=0\forall a\in\mathcal{M}\}$$. Then, every $$z\in\mathcal{H}$$ can be written $$z=x+y$$ where $$x\in\mathcal{M},y\in\mathcal{M}^{\perp}$$

Proof

Riesz representation theorem
Let $$\mathcal{H}$$ be a Hilbert space. Then, every $$\ell\in\mathcal{H}^*$$ (that is $$\ell$$ is a linear functional) can be expressed as an inner product.