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

Let be a closed real interval with , and let

endowed with pointwise addition and scalar multiplication. Setting

we obtain an inner product on . Additivity follows from

It is also positive definite: If is not the zero function, then let denote a point with . Therefore, , and due to the continuity of there exists a neighborhood of length , where

holds for some (one can take ), Hence,

is positive.