Let U , V {\displaystyle {}U,V} be vector spaces over K {\displaystyle {}K} , equipped with bilinear forms Φ U {\displaystyle {}\Phi _{U}} and Φ V {\displaystyle {}\Phi _{V}} , and let φ : U → V {\displaystyle {}\varphi \colon U\rightarrow V} denote an isometry. Is φ {\displaystyle {}\varphi } always injective?
