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.