Numerical Methods in Computational Finance. Daniel J. Duffy

Скачать книгу

      An inner product (a generalisation of dot product from high school calculus) on a real vector space V is a scalar-valued function on the Cartesian product of V with itself having the following axioms:

(5.4)

      Inner products on complex vector spaces are also possible, but a discussion is outside the scope of this chapter. We also note that inner products are sometimes written as (for example, in physics) instead of where x and y are vectors. For the specific case of n-dimensional vectors we usually write the inner product as follows:

      (5.5)

      where is the transpose of vector .

This latter notation is common in linear algebra and applications. Of course, more general vector spaces will need the more generic form in axioms (5.4).

      An inner product space is a vector space on which an inner product is defined. A finite-dimensional real inner product space is known as a Euclidean space, and a complex inner product space is known as a unitary space. The length of a vector x in Euclidean space is defined to be , and the angle between two vectors x and y is given by:

      (5.6)

      We say that two vectors x and y are orthogonal if . We immediately see that the zero vector is orthogonal to every other vector. Another example is ; then .

      Definition 5.4 The set in an inner product space is orthonormal if:

      (5.7)

      Finding an orthonormal set in an inner product space is analogous to choosing a set of mutually perpendicular unit vectors in elementary vector analysis.

      5.3.1 Orthonormal Basis

      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) Скачать книгу