User:Caliburn/s/fourier/Definition:Quasinorm on Vector Space

Definition
Let $\Bbb F$ be a subfield of $\C$.

Let $X$ be a vector space.

A quasinorm on $X$ is a map from $X$ to the nonnegative reals:


 * $\norm {\,\cdot\,} : X \to \R_{\ge 0}$

satisfying:


 * $(1) \quad$ for each $\lambda \in \Bbb F$ and $x \in X$ we have $\norm {\lambda x} = \cmod \lambda \norm x$
 * $(2) \quad$ there exists $K \ge 1$ such that $\norm {x + y} \le K \paren {\norm x + \norm y}$ for all $x, y \in X$.