Definition
A Hilbert Space is a vector space endowed with a Inner Product which is complete.Some authors will also specify that a Hilbert space must also be separable.
Relationship to Banach Spaces
Al Hilbert spaces are also Banach Spaces. In particular, the required norm is defined as
{% ||x ||^2 = \langle x | x \rangle %}
On the other hand,
not all Banach spaces can be made into Hilbert spaces.
Theorems
- Gram Schmidt - given a linearly independent set of vectors in a Hilbert space, {% (v_1,v_2, ... )_1^{\infty} %} there is an ortnormal sequence {% (u_1,u_2, ... )_1^{\infty} %} such that the span of {% u_1,u_2, ... u_k %} is the same as {% v_1,v_2, ... v_k %}
- A Hilbert space is separable iff it has a countable orthonormal basis Corollary - a subspace of a separable space has a countable orthonormal basis