Gram-Schmidt orthonormalization and orthogonalization

Load the linear algebra package (library)

<<LinearAlgebra`Orthogonalization`

Orthonormalize a typical set of independent functions (MMa treats a set as a list inside curly braces.)

Integrals range from -1 to +1 in this example - make the functions and integral ranges whatever you like (as long as the integrals exist).

functions = {1, x, x^2, x^3, x^4}

orthonormalFunctions = GramSchmidt[functions,
  InnerProduct ->
     (Integrate[#1 #2,{x,-1,1}]&)]//Simplify

Check for orthonormality

Show what the orthonormalized functions look like

Equivalent two-step process: first create orthogonal but not normalized functions from a given set of linearly independent functions, then normalize them

Check for orthonormality

2	0	0	0	0
0		0	0	0
0	0		0	0
0	0	0		0
0	0	0	0

Now normalize the orthogonal functions

normalizedFunctions = Table[Normalize[orthoFunctions[[i]],
 InnerProduct ->
     (Integrate[#1 #1,{x,-1,1}]&)],{i,1,5}]//Simplify

Check that we get the same result as simuultaneous orthogonalization and normalization