We will do many examples today to get intuition for what's Fourier transform is doing.

Let $N$ be a positive integer.

Let 'x-space' be $V_x = \Z/ N \Z = \{0,1, \cdots, N-1\}$, and then (rescaled) 'p-space' is also $V_p = \Z / N\Z$, then we can build the Fourier transformation kernel $$ K(x,p) = e^{2\pi i \frac{x p}{N}} : V_x \times V_p \to U(1) $$ where $U(1)$ is the unit circle in complex number.

The kernel satisfies the orthonormal condition in both x and p variable. $$ \frac{1}{N} \sum_{x \in V_x} K(x,p_1) \overline{K(x,p_2)} = \delta(p_1, p_2) $$ $$ \frac{1}{N} \sum_{p \in V_p} K(x_1,p) \overline{K(x_2,p)} = \delta(x_1, x_2) $$

Given a function $f(x): V_x \to \C$, we can expand it as $$ f(x) = \sum_{p \in V_p} K(x,p) F(p) $$ for some function $F(p): V_p \to \C$. Using the orthonormal condition, we get, for any $q \in V_p$, we have $$ (1/N) \sum_{x \in V_x} f(x) \overline{K(x,q)} = (1/N) \sum_{x \in V_x} \sum_{p \in V_p} K(x,p) F(p) \overline{K(x,q)} = F(q). $$ that is $$ F(q) = (1/N) \sum_{x \in V_x} f(x) \overline{K(x,q)}. $$

Let $Fun(V_x, \C)$ be the space of functions from $V_x$ to $\C$. Similarly define $Fun(V_p, \C)$. Concretely $Fun(V_x, \C) = \C^N$ and $Fun(V_p, \C) = \C^N$.

Fourier transformation $FT$, sends an element $f(x) \in Fun(V_x,\C)$ to an element $F(p) \in Fun(V_p, \C)$. $FT$ is a linear map.

If we define hermitian inner product on $Fun(V_x,\C)$ as $$ \langle f(x), g(x) \rangle_x = (1/N) \sum_{x \in V_x} f(x) \overline{g(x)}, $$ and we define hermitian inner product on $Fun(V_p,\C)$ as $$ \langle F(p), G(p) \rangle_p = \sum_{p \in V_p} F(p) \overline{G(p)}. $$ then we find that Fourier transformation is compatible with the two inner products, namely $$ \langle f, g \rangle_x = \langle F, G \rangle_p, \quad F = FT(f), G = FT(g). $$