Bases in linear algebra are extremely useful tools for analysing problems. Using a basis, one can often rephrase a complicated abstract problem in concrete terms, perhaps even suitable for a computer to work with. A basis provides a way of describing a vector space in a way that:
When translated into the language of linear algebra, we recover the key properties of a basis: that it be a spanning set and linearly independent.
In infinite dimensions, having a basis becomes more valuable as the spaces get more complicated. However, the notion of a basis also becomes complex because the question of what makes a description admits different answers depending on whether we want only finite sums, we allow sequences, or we want infinite sums.
Let be a topological vector space and a subset.
We say that is a Hamel basis if:
We say that is a topological basis if:
We say that is a Schauder basis if:
In the presence of the axiom of choice, Hamel bases always exist.
If is a topological basis, then has a dual basis. Since is not total but is total, the closure of the span of must be a codimension subspace, whence the kernel of a non-trivial continuous linear functional on , say . By scaling, this functional can be assumed to satisfy . Since , for all , .
If is a Schauder basis then it is a topological basis and so, as mentioned, has a dual basis. Then the coefficients in the sum must be given by evaluating the dual basis on : .
In with the norm :
The monomials are linearly independent and have dense span, but do not form a topological basis as there is a sequence of polynomials with no linear term converging to .
The trigonometric polynomials do form a topological basis. The dual basis is given by taking the Fourier coefficients of a function. However, it is not a Schauder basis as there are continuous functions which are not the uniform limit of their Fourier series.
The following is a Schauder basis. Let be the sequence . Define to be the piecewise-linear function with the property that: and for , and has the least “breaks”. Then forms a Schauder basis for . This is the classical Faber-Schader basis.
Enflo, P. (1973). A counterexample to the approximation problem in Banach spaces. Acta Math., 130, 309–317.
Semadeni, Z. (1982). Schauder bases in Banach spaces of continuous functions (Vol. 918). Lecture Notes in Mathematics. Berlin: Springer-Verlag.