orthogonal function and inner product space

Question

Consider the inner product space $$\langle f,g \rangle= \int_{-1}^{1} f(x) g(x) \ dx $$

find the non zero orthogonal function with respect to $f(x)=1$ in the subspace span of ${1,e^{x}}$ ?

Answer 1

If you compute the orthogonal projektion $P(e^x)$ on span(1). Then $e^x - P(e^x)$ is in the orthogonal complement of span(1).