Table of Contents

## Which is the fundamental theorem of a Hilbert space?

Fundamental Theorem of Hilbert spaces: Suppose that (H, B) is a Hausdorff pre-Hilbert space where B : H × H → ℂ is a sesquilinear form that is linear in its first coordinate and antilinear in its second coordinate.

## Which is the Hermitian form of a pre-Hilbert space?

A sesquilinear form B on H is called a Hermitian form if in addition it has the property that A pre-Hilbert space is a pair consisting of a vector space H and a non-negative sesquilinear form B on H; if in addition this sesquilinear form B is positive definite then (H, B) is called a Hausdorff pre-Hilbert space.

## Which is the Hausdorff form of a pre-Hilbert space?

A pre-Hilbert space is a pair consisting of a vector space H and a non-negative sesquilinear form B on H; if in addition this sesquilinear form B is positive definite then (H, B) is called a Hausdorff pre-Hilbert space. If B is non-negative then it induces a canonical seminorm on H, denoted by

## How are compact operators closed in a Hilbert space?

For any separable Hilbert space, the compact operators form a closed and -closed two-sided ideal in B(H): Proof. In any metric space (applied to B(H)) the closure of a set is closed, so the compact operators are closed being the closure of the nite rank operators.

## Which is the reproducing Hilbert space in functional analysis?

Reproducing kernel Hilbert space. In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional.

## Is the L2 space a Hilbert space of a function?

Note that L2 spaces are not Hilbert spaces of functions (and hence not RKHSs), but rather Hilbert spaces of equivalence classes of functions (for example, the functions

## Why is geometric intuition important in Hilbert space?

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space.

## When to use fundamental and intermediate representations and warranties?

Use of fundamental and intermediate representations and warranties, when done effectively, can help the parties strike a balance between the seller’s and buyer’s risks and goals. If you have any questions about fundamental representations and warranties, survival periods, or caps, please contact Jon Siebers.