Definition:Orthonormal Subset

Definition

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

Let $S \subseteq V$ be a subset of $V$.

Then $S$ is an orthonormal subset (of $V$) if and only if:

$(1): \quad \forall u \in S: \norm u = 1$

where $\norm {\, \cdot \,}$ is the inner product norm.

$(2): \quad S$ is an orthogonal set:

$\forall u, v \in S: u \ne v \implies \innerprod u v = 0$

Examples

The $L^2$ Space $L^2_\C \closedint 0 {2 \pi}$

Let $L^2_\C \closedint 0 {2 \pi}$ be the complex $L^2$ space over the closed interval $\closedint 0 {2 \pi}$.

Let $\innerprod \cdot \cdot$ be the $L^2$ inner product.

For $n \in \Z$, let $e_n: \closedint 0 {2 \pi} \to \C$ be defined by:

$\map {e_n} t = \paren{ 2 \pi }^{-1/2} \map \exp {i n t}$

Then $\set{ e_n : n \in \Z}$ is an orthonormal subset of $L^2_\C \closedint 0 {2 \pi}$.

Also see

• Definition:Basis (Hilbert Space)
• Orthonormal Subset of Hilbert Space Extends to Basis

• Results about orthonormal subsets can be found here.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 4.$ Orthonormal Sets of Vectors and Bases: Definition $4.1$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): orthonormal
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): orthonormal
• 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) $9.2$: Orthonormal Sets