Definition:Hilbert Space

Definition
Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$.

Notation
The subscripts of the inner product $\innerprod \cdot \cdot_H$ and the inner product norm $\norm {\,\cdot\,}_H$ on $H$ serve to emphasize the space $H$ when considering multiple Hilbert spaces.

If it is clear from context which Hilbert space we are studying, the subscripts may be omitted such that::


 * $\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_H$ will denote the inner product on $H$
 * $\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_H$ will denote the inner product norm on $H$

Make sure to understand the precise definition of (especially) the inner product.

Also see

 * Equivalence of Definitions of Hilbert Space
 * Hilbert Space is Hausdorff Topological Vector Space
 * Definition:Banach Space
 * Definition:Linear Functional, for more discussion on notation.
 * Definition:Linear Transformation, for more discussion on notation.