Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 83

Let X be an inner product space. The set is orthonormal if for and for . The set is called an orthonormal basis.

Then .

The inner product space (continuous functions) has:

 Orthnormal basis .

 Orthogonality because .

An interesting application of inner products is to kernel theory to statistical learning in Learning with Kernels, Schölkopf and Smola (2002). In this case we do not work in an original (let's say n-dimensional) space X but in a feature space H. To this end, consider the map:

(5.8)

We embed data into H, and this approach offers several advantages, one of which is that we can define a similarity measure from the inner product in H:

(5.9)

The function is called a kernel.