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

## 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.