Definition:Hilbert Space

Definition
Let $V$ be an inner product space over $\Bbb F \in \set {\R, \C}$.

Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_V$.

If $\struct {V, d}$ is a complete metric space, $V$ is said to be a Hilbert space.

The Hilbert space $V$ may be considered as one of the following:


 * The complete inner product space $\struct {V, \innerprod \cdot \cdot_V}$
 * The Banach space $\struct {V, \norm {\,\cdot\,}_V}$
 * The topological space $\struct {V, \tau_d}$ where $\tau_d$ is the topology induced by $d$
 * The vector space $\struct {V, +, \circ}_{\Bbb F}$

That is to say, all theorems and definitions for these types of spaces directly carry over to all Hilbert spaces.

Standard Notation
In most of the literature, when studying a Hilbert space $\HH$, unless specified otherwise, it is understood that:


 * $\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_\HH$ denotes the inner product on $\HH$
 * $\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_\HH$ denotes the inner product norm on $\HH$

where the subscripts serve to emphasize the space $\HH$ when considering multiple Hilbert spaces.

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

Furthermore, the parentheses around the argument of linear functionals and linear transformations on $\HH$ are often suppressed for brevity.

Make sure to understand which symbols denote scalars, operators and functionals, respectively.

Also see

 * Definition:Banach Space