Introducing Inner Product Spaces: Real-World Examples

[matqkks]

What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inner product of general vectors spaces such as the set of matrices, polynomials, functions?

 

[Ackbach]

What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inner product of general vectors spaces such as the set of matrices, polynomials, functions?

You can't do better than quantum mechanics and Hilbert space, which is a particular kind of inner product space. If you look at the hydrogen atom, for example, the Schrödinger equation is an eigenvalue problem. The eigenvectors are functions that live in a Hilbert space. You can use the Gram-Schmidt process to reduce the eigenvectors to an orthonormal basis, which is probably the most useful kind of basis. The Gram-Schmidt process makes use of the inner product extensively.