None 2023
The properties that an inner product on an abstract vector space must satisfy can be found in:
Definition and Properties of an Inner Product.
Definition: The inner product for any two vectors in the vector space of
periodic functions with a given period (let's pick \(2\pi\) for simplicity)
is given by:
\[\left\langle {f}\middle|{g}\right\rangle =\int_0^{2\pi} f^*(x)\, g(x)\, dx\]

Show that the first property of inner products
\[\left\langle {f}\middle|{g}\right\rangle =\left\langle {g}\middle|{f}\right\rangle ^*\]
is satisfied for this definition.

Show that the second property of inner products
\[\left\langle {f}\right|\Big(\lambda\left|{g}\right\rangle + \mu \left|{h}\right\rangle \Big) = \lambda\left\langle {f}\middle|{g}\right\rangle +\mu\left\langle {f}\middle|{h}\right\rangle \]
is satisfied for this definition.