# Mathematical Methods of Physics/Riesz representation theorem

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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.