Semi-Inner Product/Examples
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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 $V$.