Definition:Semi-Inner Product
Definition
Complex Semi-Inner Product
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 | \(\ds \forall x \in V:\) | \(\ds \quad \innerprod x x \in \R_{\ge 0} \) |
Real Semi-Inner Product
Let $V$ be a vector space over a real subfield $\GF$.
A (real) semi-inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the real semi-inner product axioms:
| \((1^\prime)\) | $:$ | Symmetry | \(\ds \forall x, y \in V:\) | \(\ds \innerprod x y = \innerprod y x \) | |||||
| \((2)\) | $:$ | 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 | \(\ds \forall x \in V:\) | \(\ds \quad \innerprod x x \in \R_{\ge 0} \) |
Semi-Inner Product Space
A semi-inner product space is a vector space together with an associated semi-inner product.
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: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 $1.1$