# 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.

## 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

## 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.