Definition 10.2.1 (Pseudonorm).label Let $E$ be a vector space over $K \in \RC$, then a pseudonorm is a function $\rho: E \to [0, \infty)$ such that

1. (PN1)

   $\rho(0) = 0$.

2. (PN2)

   For any $x \in X$ and $\lambda \in K$ with $\abs{\lambda}\le 1$, $\rho(\lambda x) \le \rho(x)$.

3. (PN3)

   For any $x, y \in X$, $\rho(x + y) \le \rho(x) + \rho(y)$.

4. (PN4)

   For any $x \in X$, $\lim_{\lambda \to 0}\rho(\lambda x) = 0$.

5. (PN5)

   For any $\lambda \in K$, and $\seq{x_n}\subset X$ with $\rho(x_{n}) \to 0$ as $n \to \infty$, $\limv{n}\rho(\lambda x_{n}) = 0$.