Definition
Hilbert Space
A Hilbert space
is a real or complex inner product space whose induced metric space is complete.
Construction
A Hilbert space has three layers of structure.
- It is a real or complex vector space
. - It has an inner product
. - The inner product induces a norm
This norm induces the inner-product-induced metric space with metric
The space is Hilbert exactly when this inner-product-induced metric space is complete.