Definition:Hilbert Space/Notation

Hilbert Space: Notation

The subscripts of the inner product $\innerprod \cdot \cdot_H$ and the inner product norm $\norm {\,\cdot\,}_H$ on $H$ serve to emphasize the space $H$ when considering multiple Hilbert spaces.

If it is clear from context which Hilbert space we are studying, the subscripts may be omitted such that::

$\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_H$ will denote the inner product on $H$

$\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_H$ will denote the inner product norm on $H$

Make sure to understand the precise definition of (especially) the inner product.

Also see

• Results about Hilbert spaces can be found here.

Source of Name

This entry was named for David Hilbert.

Sources

• 1997: Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis: $\S 11$: Hilbert Spaces