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.