View source for Mathematical Methods of Physics/Riesz representation theorem

In this chapter, we will more formally discuss the bra <math>|\rangle</math> and ket <math>\langle |</math> notation introduced in the previous chapter.

==Projections==
===Definition===
Let <math>\mathcal{H}</math> be a Hilbert space and let <math>\ell:\mathcal{H}\to\mathbb{C}</math> be a continuous linear transformation. Then <math>\ell</math> is said to be a '''linear functional''' on <math>\mathcal{H}</math>.

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

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

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

'''Proof'''


==Riesz representation theorem==
Let <math>\mathcal{H}</math> be a Hilbert space. Then, every <math>\ell\in\mathcal{H}^*</math> (that is <math>\ell</math> is a linear functional) can be expressed as an inner product.

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