Mathematical Methods of Physics/Reisz representation theorem

< Mathematical Methods of Physics

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

Contents

ProjectionsEdit

DefinitionEdit

Let   be a Hilbert space and let   be a continuous linear transformation. Then   is said to be a linear functional on  .

The space of all linear functionals on   is denoted as  . Notice that   is a normed vector space on   with  

We also have the obvious definition,   are said to be orthogonal if  . We write this as  . If   then we write   if  

TheoremEdit

Let   be a Hilbert space, let   be a closed subspace of   and let  . Then, every   can be written   where  

Proof


Reisz representation theoremEdit

Let   be a Hilbert space. Then, every   (that is   is a linear functional) can be expressed as an inner product.