Mathematical Methods of Physics/Riesz representation theorem

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

Projections

Definition

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

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

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

Theorem

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

Proof

Riesz representation theorem

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