The inner product of two vectors $\vec a$ and $\vec b$ of n dimensions, is given by,
$$(\vec a \,, \vec{b}) =a_1b_1 + a_2b_2 + a_3b_3 + \,\,…\,\,+a_nb_n $$
If a function is considered to be a vector of infinite dimensions, for example,
$$f(x) = \begin{bmatrix} \vdots \\ f(0.05) \\ f(0.051) \\ f(0.052) \\ \vdots \end{bmatrix}$$
Then, the inner product of two functions say $f(x), g(x)$ is given by $$\mathbf {\bigl(} f(x),g(x)\mathbf {\bigr)}=\int_{-\infty}^\infty f(x)g(x) \,\,\, \mathbf{dx}$$
My Question is:
How did we get the $\mathbf {dx}$ in the inner product (or)$$ \mathbf{why\,\, is}\,\,\,\,\ \sum_{-\infty}^\infty f(x)g(x) = \int_{-\infty}^\infty f(x)g(x) \,\,\, \mathbf{dx}$$
In a Riemann sum , the $\Delta x $ term becomes $dx$ during the limiting process, but in this sum, there doesn't seem to be a $\Delta x$ term. I think the answer to this question would probably answer my another question,
Best Answer
The $\Delta x$ term is just $1$, as the sum is over the integers. As you shorten the range, it is clear that a $\Delta x$ term is necessary.