Seminorm Maps Zero Vector to Zero
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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 | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\blacksquare$