Definition:Inner Product Norm

Definition

Let $\Bbb F$ be a subfield of $\C$.

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\Bbb F$.

Then the inner product norm on $V$ is the mapping $\norm {\, \cdot \,} : V \to \R_{\ge 0}$ given by:

$\norm x = \sqrt {\innerprod x x}$

for each $x \in V$.

Also see

• Inner Product Norm is Norm

• Definition:Hilbert Space
• Definition:Banach Space

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): parallelogram law: 1.
• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): norm: 1. (of a vector space)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): norm: 1. (of a vector space)
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $14.1$: The Cauchy-Schwarz Inequality