Inner Product of Functions

The inner/dot product for Euclidean vectors is defined as: $$ \langle u, v \rangle = \sum_{i} u_i v_i $$ Following on from how we defined a norm for square integrable functions , we can similarly define an inner product for such functions in $\mathbb{L}^2$: $$ \langle f, g \rangle = \int_{-\infty}^{\infty} f(x) g(x) dx $$ This inner product can now be used to define a norm /length for our space:…