Definition:Inner Product Norm

Definition

Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$.

Let $\left \langle{\cdot, \cdot}\right \rangle$ be the inner product of $V$.

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

$\left\Vert{x}\right\Vert := \left\langle{x,x}\right\rangle^{1/2}$.

Also see

• Inner Product Norm is Norm

• Definition:Hilbert Space
• Definition:Banach Space

Sources

• 1990: John B. Conway: A Course in Functional Analysis $\S I.1$