Symmetry of Invariant Metric on Vector Space

Theorem

Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $d$ be an invariant metric on $X$.

Then we have:

$\map d {x, y} = \map d {-x, -y}$

for each $x, y \in X$.

Proof

Let $x, y \in X$.

We have:

\(\ds \map d {x, y}\)

\(=\)

\(\ds \map d {x + \paren {-y - x}, y + \paren {-y - x} }\)

Definition of Invariant Metric on Vector Space

\(\ds \)

\(=\)

\(\ds \map d {-y, -x}\)

\(\ds \)

\(=\)

\(\ds \map d {-x, -y}\)

Metric Space Axiom $(\text M 3)$

$\blacksquare$