Last modified on 31 March 2010, at 08:12

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.

ProjectionsEdit

DefinitionEdit

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

TheoremEdit

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 theoremEdit

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.