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 &phi;(x) as:


 * $$\phi_n(x) = \left\{\begin{matrix}0 & x \le 0 \\

nx & 0 < x \le \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 &phi; converges to a function &phi;* 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.