Definition:Semi-Inner Product/Complex Field
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Definition
Let $V$ be a vector space over a complex subfield $\GF$.
A (complex) semi-inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the (complex) semi-inner product axioms:
\((1)\) | $:$ | Conjugate Symmetry | \(\ds \forall x, y \in V:\) | \(\ds \quad \innerprod x y = \overline {\innerprod y x} \) | |||||
\((2)\) | $:$ | Sesquilinearity | \(\ds \forall x, y, z \in V, \forall a \in \GF:\) | \(\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z \) | |||||
\((3)\) | $:$ | Non-Negative Definiteness | \(\ds \forall x \in V:\) | \(\ds \quad \innerprod x x \in \R_{\ge 0} \) |
Complex Semi-Inner Product Space
Let $V$ be a vector space over a complex subfield $\GF$.
Let $\innerprod \cdot \cdot : V \times V \to \GF$ be an complex semi-inner product on $V$.
We say that $\struct {V, \innerprod \cdot \cdot}$ is a (complex) semi-inner product space.
Examples
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 $V$.
Also see
- Definition:Complex Inner Product, a semi-inner product with the additional property of positiveness.
Sources
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- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Definition $1.1$