Definition:Hilbert Space/Definition 1

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

• Equivalence of Definitions of Hilbert Space

• 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