nLab
basis in functional analysis

Concepts of basis in functional analysis

Idea

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:

  1. Is complete: every point in the space can be described in this fashion.
  2. Has no redundancies: the description of a point is unique.

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.

Definitions

Let VV be a topological vector space and BVB \subseteq V a subset.

  1. We say that BB is a Hamel basis if:

    1. Every element of vv is a finite linear combination of elements of BB,
    2. If v= bBα bbv = \sum_{b \in B} \alpha_b b then the α b\alpha_b are unique.

    Alternatively, BB is linearly independent and Span(B)=V\Span(B) = V; in other words, the span of BB is VV but no proper subset of BB has this property.

  2. We say that BB is a topological basis if:

    1. Every element vVv \in V is a limit of a sequence or (more generally) a net of finite linear combinations of elements of BB,
    2. No element of BB is a limit of a sequence or net of finite linear combinations of the other elements of BB.

    Alternatively, BB is total? (meaning that its span is dense) but no proper subset of BB is total.

  3. We say that BB is a Schauder basis if:

    1. Every element of vv is a (possibly infinite) sum of scales of elements of BB,
    2. If v= bBα bbv = \sum_{b \in B} \alpha_b b then the α b\alpha_b are unique.

Properties

  1. In the presence of the axiom of choice, Hamel bases always exist.

  2. If BB is a topological basis, then BB has a dual basis. Since B{b}B \setminus \{b\} is not total but BB is total, the closure of the span of B{b}B \setminus \{b\} must be a codimension 11 subspace, whence the kernel of a non-trivial continuous linear functional on VV, say f bf_b. By scaling, this functional can be assumed to satisfy f b(b)=1f_b(b) = 1. Since B{b}kerfB \setminus \{b\} \subseteq \ker f, f(b)=0f(b') = 0 for all bBb' \in B, bbb' \ne b.

  3. If BB is a Schauder basis then it is a topological basis and so, as mentioned, has a dual basis. Then the coefficients in the sum v=α bbv = \sum \alpha_b b must be given by evaluating the dual basis on vv: v=f b(v)bv = \sum f_b(v) b.

Examples

  1. In C([0,1],)C([0,1],\mathbb{C}) with the norm f=max{|f(t)|}{\|f\|} = \max\{{|f(t)|}\}:

    1. 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 tt.

    2. 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.

    3. The following is a Schauder basis. Let (d n)(d_n) be the sequence {0,1,12,14,34,}\{0, 1, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \dots\}. Define f nf_n to be the piecewise-linear function with the property that: f n(d n)=1f_n(d_n) = 1 and f n(d k)=0f_n(d_k) = 0 for k<nk \lt n, and f nf_n has the least “breaks”. Then f nf_n forms a Schauder basis for C([0,1],)C([0,1],\mathbb{C}). This is the classical Faber-Schader basis.

References

  • 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.

Revised on August 16, 2012 00:00:49 by Toby Bartels (98.19.44.121)