Linear mapping/Bilinear form/Isometry/Definition

Isometry (bilinear form)

Let $V$ and $W$ be vector spaces over $K$, both endowed with a bilinear form ${\mathrm{\Phi }}_V$ and ${\mathrm{\Phi }}_W$, A $K$-linear mapping $f:V\to W$ is called an isometry if

${\mathrm{\Phi }}_W(f(u),f(v))={\mathrm{\Phi }}_V(u,v)$

holds for all $u,v\in V$.