Continuous functions/Interval/C-valued/Inner product/Example

Let ${}[a,b]$ be a closed real interval with ${}a<b$, and let

${}V={\left\{f:[a,b]\rightarrow \mathbb {C} \mid f{\text{ continuous}}\right\}}\,,$

endowed with pointwise addition and scalar multiplication. Setting

${}\left\langle f,g\right\rangle :=\int _{a}^{b}f(t){\overline {g(t)}}dt\,,$

we obtain an inner product on ${}V$. Additivity follows from

${}\left\langle f_{1}+f_{2},g\right\rangle =\int _{a}^{b}(f_{1}(t)+f_{2}(t)){\overline {g(t)}}dt=\int _{a}^{b}f_{1}(t){\overline {g(t)}}dt+\int _{a}^{b}f_{2}(t){\overline {g(t)}}dt=\left\langle f_{1},g\right\rangle +\left\langle f_{2},g\right\rangle \,.$

It is also positive definite: If ${}f$ is not the zero function, then let ${}s\in [a,b]$ denote a point with ${}f(s)\neq 0$. Therefore, ${}\vert {f(s)}\vert >0$, and due to the continuity of ${}f$, there exists a neighborhood ${}[s-\epsilon ,s+\epsilon ]\cap [a,b]$ of length ${}\geq \epsilon$ where

${}\vert {f(t)}\vert \geq c>0\,$

holds for some ${}c\in \mathbb {R} _{+}$ (one can take ${}c={\frac {\vert {f(s)}\vert }{2}}$). Hence,

${}\left\langle f,f\right\rangle =\int _{a}^{b}\vert {f(t)}\vert ^{2}dt\geq \int _{\max {\left(a,s-\epsilon \right)}}^{\min {\left(b,s+\epsilon \right)}}\vert {f(t)}\vert ^{2}dt\geq \epsilon c^{2}>0\,$

is positive.