Definition:Hilbert-Schmidt Norm

Definition

Let $A = \paren {a_{i j} }_{m \times n} \in \R^{m \times n}$ be an $m \times n$ matrix.

Let $\norm {\, \cdot \,}_{HS} : \R^{m \times n} \to \R$ be a mapping such that:

$\ds \norm A_{HS} = \sqrt {\sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^m a_{ij}^2 }$

Then $\norm {\, \cdot \,}_{HS}$ is called the Hilbert-Schmidt norm.

Source of Name

This entry was named for David Hilbert and Erhard Schmidt.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$. Operator norm and the normed space $\map {CL} {X, Y}$