3.7 Inner Product of Functions
The concept of inner product extends from finite-dimensional vectors to functions.
Definition 3.10 For functions \(u\) and \(v\), we define the inner product of \(u\) and \(v\) as: \[ \langle u, v \rangle = \int_a^b u(x)v(x) \, dx. \]
Example 3.18 Consider the space of continuous functions on \([0,1]\), and define the inner product \[ \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt. \]
Let \(f(t) = t\) and \(g(t) = t^2\). Then
\[ \langle f, g \rangle = \int_0^1 t \cdot t^2 \, dt = \int_0^1 t^3 \, dt = \frac{1}{4}. \]
If \(\langle u, v \rangle = 0\), the functions are orthogonal.
Example 3.19 Using this inner product, \(\sin(x)\) and \(\cos(x)\) are orthogonal on \([-\pi, \pi]\).
Any collection of functions from the set \(\{1, \cos(x), \cos(2x), \ldots\}\) also forms an orthogonal system on this interval.