Numerical Methods in Computational Finance. Daniel J. Duffy

      A space X endowed with a metric d is called a metric space and is denoted by (X, d).

      Examples of metrics are:

      1 

      2 Let X be a non-empty set(4.9)

      1 Let X be a set and let be the set of p-integrable Lebesgue functions on X. If , then a metric is:

      Norms and metrics are important quantities when proving convergence results in functional and numerical analysis applications.

4.3 SUBSPACES

      A non-empty subset X of a vector space ) is called a vector subspace of if X forms a vector space over K with the same addition and scalar multiplication as in . For example, let P be the set of polynomials in X with real coefficients, and let polynomial addition and multiplication by real numbers be defined by:

      (4.10)

Now let be the set of consisting of all polynomials of at most degree of the form:

(4.11)

      Then is a subspace of P, and it is also a subspace of .

      We say that a subset X of a vector space is said to be closed under addition if whenever , then . A subset X of a vector space is said to be closed under scalar multiplication if whenever and then .

      Theorem 4.1 A subset X of a vector space is a subspace if and only if:

      (4.12)

      An exercise: let be any r elements of a vector space . Prove that the set U of all elements of that can be written in the form forms a subspace of .

      We give an example of a subset X of defined by:

      (4.13)

      It is easily verified that X is a vector space over K, but X is not a subspace of because:

      and these two quantities are thus not the same!

4.4 LINEAR INDEPENDENCE AND BASES

      We are now interested in finding a minimal subspace U of independent vectors containing a set X (U contains X) such that any vector in X can be written as a linear combination of these vectors. In this case we say that X spans U. We are particularly interested in the case , that is subsets of that span