IEVref: 103-01-08 ID: Language: en Status: Standard Term: inner product Synonym1: Synonym2: Synonym3: Symbol: Definition: for two complex-valued functions, $\underset{_}{f}$ and $\underset{_}{g}$, defined on the interval $\left[a,b\right]$ of ℝ, complex number $〈\underset{_}{f},\underset{_}{g}〉={\int }_{a}^{b}\underset{_}{f}\left(x\right){\underset{_}{g}}^{\ast }\left(x\right)\text{d}x$, where ${\underset{_}{g}}^{\ast }$ is the conjugate of $\underset{_}{g}$Note 1 to entry: The inner product has the following properties: $〈\underset{_}{f},\underset{_}{g}〉={〈\underset{_}{g},\underset{_}{f}〉}^{\ast }$ and $〈\underset{_}{\alpha }\text{\hspace{0.17em}}\underset{_}{f}+\underset{_}{\beta }\text{\hspace{0.17em}}\underset{_}{g},\underset{_}{h}〉=\underset{_}{\alpha }〈\underset{_}{f},\underset{_}{h}〉+\underset{_}{\beta }〈\underset{_}{g},\underset{_}{h}〉$ where $\underset{_}{\alpha },\underset{_}{\beta }\in$ ℂ. Note 2 to entry: The inner product for complex functions is similar to the Hermitian product for vectors (see IEC 60050-102, 102-03-18). For real functions, it is similar to the scalar product (see IEC 60050-102, 102-03-17). Publication date: 2009-12 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00020 (IEV 103) - evaluation. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: