Semi-Inner Product/Examples

Examples of Semi-Inner Products

Sequences with Finite Support

Let $\GF$ be a subfield of $\C$.

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

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

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

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