Engineering Analysis/Banach and Hilbert Spaces

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

Convergent Functions

Let's define the piece-wise function φ(x) as:

$\phi _{n}(x)=\left\{{\begin{matrix}0&x\leq 0\\nx&0<x\leq {\frac {1}{n}}\\1&{\frac {1}{n}}<x\end{matrix}}\right.$

We can see that as we set $n\to \infty$, 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 L₂ space, because the unit step function does not exist in the C space (it is not continuous).

Convergence

We can say that a function φ converges to a function φ^* if:

$\lim _{n\to \infty }\|\phi _{n}-\phi ^{*}\|=0$

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

A function space is called complete if all sequences in that space converge to another function in that space.

Banach Space

A Banach Space is a complete normed function space.

Hilbert Space

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:

$\|f\|=g(\langle f,f\rangle )$

That is, that the norm can be written as a function of the scalar product. In the L₂ space, we can define the norm as:

$\|f\|={\sqrt {\langle f,f\rangle }}$

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:

$\|f+g\|^{2}+\|f-g\|^{2}=2\|f\|^{2}+2\|g\|^{2}$

The L₂ space is a Hilbert Space. The C space, however, is not.