# Engineering Analysis/Banach and Hilbert Spaces

There are some special spaces known as **Banach** spaces, and **Hilbert** spaces.

## Convergent Functions

[edit | edit source]Let's define the piece-wise function φ(x) as:

We can see that as we set , this function becomes the unit step function. We can say that as n approaches infinity, that this function converges to the unit step function. Notice that this function only converges in the L_{2} space, because the unit step function does not exist in the C space (it is not continuous).

### Convergence

[edit | edit source]We can say that a function φ converges to a function φ^{*} if:

We can call this sequences, and all such sequences that converge to a given function as n approaches infinity a **cauchy sequence**.

### Complete Function Spaces

[edit | edit source]A function space is called complete if all sequences in that space converge to another function in that space.

## Banach Space

[edit | edit source]A Banach Space is a complete normed function space.

## Hilbert Space

[edit | edit source]A Hilbert Space is a Banach Space with respect to a norm induced by the scalar product. That is, if there is a scalar product in the space X, then we can say the norm is induced by the scalar product if we can write:

That is, that the norm can be written as a function of the scalar product. In the L_{2} space, we can define the norm as:

If the scalar product space is a Banach Space, if the norm space is also a Banach space.

In a Hilbert Space, the Parallelogram rule holds for all members f and g in the function space:

The L_{2} space is a Hilbert Space. The C space, however, is not.