Mathematical Methods of Physics/Reisz representation theorem

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

Projections

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

Reisz 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.