The proof of this is similar to that of Lemma X above and Lemma X below. \end{iexample} Example: Given vector spaces $ U $ , $ V $ and $ W $ , a bilinear map $ f{\colon}\linebreak[0] U \times V \to W $ is a function $ f $ that is linear in each variable: $ f(u, v_1 + \lambda v_2) &= f(u, v_1) + \lambda f(u, v_2), \ f(u_1 + \lambda u_2, v) &= f(u_1, v) + \lambda f(u_2, v) $ for all $ u, u_1, u_2 \in U $ , $ v, v_1, v_2 \in V $ , and scalars $ \lambda $ . A good example is the scalar product (dot product), which is a bilinear map $ \begin{array}{ccc} \reals^n \times \reals^n &\to &\reals \ (\mathbf{u}, \mathbf{v}) &\mapsto &\mathbf{u}.\mathbf{v} \end{array} $ of real vector spaces.