Seminorm Maps Zero Vector to Zero

Theorem

Let $\struct {K, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_K$.

Let $X$ be a vector space over $\struct {K, \norm {\,\cdot\,}_K}$.

Let $\mathbf 0_X$ be the zero vector of $X$.

Let $p$ be a seminorm on $X$.

Then $\map p {\mathbf 0_X} = 0$.

Proof

We have:

\(\ds \map p {\mathbf 0_X}\)

\(=\)

\(\ds \map p {0 \circ \mathbf 0_X}\)

Zero Vector Scaled is Zero Vector

\(\ds \)

\(=\)

\(\ds \norm 0_K \map p {\mathbf 0_X}\)

$(\text N 2)$ in Definition of Seminorm

\(\ds \)

\(=\)

\(\ds 0 \; \map p {\mathbf 0_X}\)

$(\text N 1)$ in Definition of Norm on Division Ring

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$