Definition:Hilbert Space/Definition 1
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Definition
Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$.
Let $\struct {H, \innerprod \cdot \cdot_H}$ be an inner product space.
Let $d: H \times H \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_H$.
Let $\struct {H, d}$ be a complete metric space.
Then $H$ is a Hilbert space over $\mathbb F$.
Also see
- Results about Hilbert spaces can be found here.
Source of Name
This entry was named for David Hilbert.
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Definition $1.6$
- 1997: Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis: $\S 11$: Hilbert Spaces
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hilbert space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hilbert space