# Mathematical Methods of Physics/Riesz representation theorem

(Redirected from Mathematical Methods of Physics/Reisz representation theorem)

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

## ProjectionsEdit

### DefinitionEdit

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

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

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

### TheoremEdit

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

Proof

## Riesz representation theoremEdit

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