What's a Normalized function?

Question

I'm studing Representations of Random Processes and the book talks about Orthonormal functions, but doesn't make it clear what is it.

I was able to realize that a set of functions are orthonormal if they are normalized and orthogonal.

I'm familiar with the concept of orthogonal functions, but I couldn't find out what exactly is a normalized function in that case, since is a widely used term in math.

I'm studing using the following book: Detection, Estimation, and Linear Modulation Theory Part 1 by Harry L. Van Trees

Answer 1

Similar to a a vector that is normal when it's magnitude is 1, which is the inner product with itself. In the same fashion, a function is normal over a defined range $[t_1,t_2]$ when the root of the integrated squared magnitude is one (if it's normal the root is trivial but this shows the same form as derived from an inner product):

$$||x(t)|| = \sqrt{\int_{t_1}^{t_2}|x(t)|^2dt} = 1 $$

Normalization of functions is done with a simple scaling: If the above formula results in a number of than one, then the entire function is scaled by that number.

Answer 2

Orthogonal means for two (real) vectors $u$ and $v$ that the scalar product vanishes: $$ \lt \overline{u},v\gt = 0$$ Under this definition, any $0$ vector is trivially orthogonal to others. As the scalar product is bilinear (sesquilinear in the complex form), one may "normalize" non-zero vectors to unit norms (induced by the scalar product: $\|u\|^2=\lt \overline{u},u\gt $):

$$ \|u\|=\|v\|=1$$

The difficulty in some texts is (in my experience) that some people write orthogonal assuming non-zero vectors, and some other use orthogonal instead of orthonormal assuming unit-normed non-zero vectors.

Additionally, note that normalizing has a different meaning: a probability function can be normalized inot a probability density function, with total probability of one