Mathematical Methods of Physics/Riesz representation theorem

From Wikibooks, open books for an open world
Jump to navigation Jump to search

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

Projections[edit]

Definition[edit]

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

Theorem[edit]

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

Proof


Riesz representation theorem[edit]

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