Semi-Inner Product/Examples/Sequences with Finite Support

Example of Semi-Inner Product
Let $\F$ be a subfield of $\C$.

Let $V$ be the vector space of sequences with finite support over $\F$.

Let $\innerprod \cdot \cdot: V \times V \to \F$ be the mapping defined by:


 * $\ds \innerprod {\sequence{a_n} } {\sequence{b_n} } = \sum_{n \mathop = 1}^\infty a_{2n} \overline{ b_{2n} }$

Then $\innerprod \cdot \cdot$ is a semi-inner product on $V$ but not an inner product on $V$.