Schauder basis

A Schauder basis of a Banach space $X$ is a sequence $(e_n)$ such that every $x\in X$ has a unique norm-convergent expansion $x={\sum }_{n=1}^{\infty }{a}_n{e}_n$.

Relative to a Schauder basis $(e_j)$, a block basic sequence consists of nonzero finite combinations supported on successive disjoint coordinate intervals. In the l-p sequence space, disjointly supported vectors $u_n$ of norm one satisfy $\Vert \sum a_n{u}_n{\Vert }_p^p=\sum |a_n{|}^p$. Their closed span is consequently isometric to the l-p sequence space.

Suppose $(u_n)$ is a normalized block basic sequence in the l-p sequence space, $1\le p<\infty$, and ${\sum }_n\Vert x_n-u_n{\Vert }_p<1$. Choose norm-one linear functionals ${\varphi }_n$ supported on the respective blocks with ${\varphi }_n(u_n)=1$. The bounded linear operator $A=I+{\sum }_n{\varphi }_n(\cdot )(x_n-u_n)$ is invertible by the Neumann series. The block projection $Qz={\sum }_n{\varphi }_n(z)u_n$ has norm at most one, and $AQ{A}^{-1}$ projects onto the closed span of $(x_n)$. Thus that span is a complemented subspace isomorphic to the l-p sequence space.

Relative to a Schauder basis, a block sequence consists of nonzero finitely supported vectors with successive disjoint supports: $\max \mathrm{supp}{x}_i<\min \mathrm{supp}{x}_{i+1}$. It is a basic sequence, with basis constant at most that of the original basis.

A block subspace is the closed linear span of an infinite block sequence. Passing to a block subspace preserves the order of supports and permits successive-vector constructions.

If an analytic family of finite normalized block sequences meets every infinite-dimensional block subspace, then after an arbitrarily prescribed coordinatewise perturbation it is strategically large in some block subspace. In the block game, one player chooses an infinite-dimensional block subspace at each turn and the other chooses a successive unit vector from it; strategically large means the vector player can force a finite initial segment into the family.

The nth basis projection associated with a Schauder basis is

The basis constant of a Schauder basis is ${\sup }_n\Vert P_n\Vert$, where $P_n$ are its basis projections. Uniform boundedness makes this supremum finite.

The nth coordinate functional of a Schauder basis is the bounded linear functional $e_n^{\ast }({\sum }_k{a}_k{e}_k)=a_n$.

The dual sequence consists of the coordinate functionals $(e_n^{\ast })$ in $X^{\ast }$. It is always a basic sequence, though it need not span the entire dual space.

A Schauder basis is shrinking when its dual sequence is a Schauder basis of the whole dual space. Every Schauder basis of a reflexive Banach space is shrinking.

A basic sequence in a Banach space is a sequence that is a Schauder basis for its closed linear span.

Every bounded sequence in a Banach space has either a weakly Cauchy sequence as a subsequence or a subsequence equivalent to the unit vector basis of ${\ell }^1$. The real-scalar proof represents vectors as continuous functions on a compact dual ball and uses infinite Ramsey theory to select independent lower and upper level sets when pointwise convergence cannot be obtained.

For $K\ge 1$, let $T_K(X)$ consist of finite normalized sequences satisfying $K^{-1}\sum |a_i|\le \Vert \sum a_i{x}_i\Vert \le \sum |a_i|$ for every scalar combination. Its tree order measures the depth of finite copies of the unit vector basis of ${\ell }^1$. A separable Banach space containing no copy of ${\ell }^1$ has a countable index because every such tree is closed and well-founded. Countable well-founded trees can be used to construct separable reflexive spaces with arbitrarily high index, precluding a separable reflexive universal space.

An unconditional basic sequence is a Schauder basis for its closed span whose expansions converge after arbitrary permutations or coordinate suppressions. Equivalently, the finite coordinate projections are uniformly bounded. Failure of this property gives pairs of disjointly supported unit vectors at arbitrarily small distance, obtained by separating a finite combination with a large coordinate projection.

Let $T_C(X)$ consist of finite normalized sequences for which every scalar combination and sign change satisfy $\Vert {\sum }_i{\epsilon }_i{a}_i{x}_i\Vert \le C\Vert {\sum }_i{a}_i{x}_i\Vert$. These inequalities give a closed tree on a Polish space when $X$ is separable. An infinite branch is an unconditional basic sequence. If there is no such sequence, each fixed-constant tree is well-founded and has countable rank of a well-founded tree. Consequently, finite unconditional configurations with one fixed constant and arbitrarily high countable ranks force an infinite unconditional sequence.

A basic sequence is suppression-unconditional with constant $C$ if every coordinate deletion satisfies the displayed inequality. When $C=1$, deletion never increases the norm. Uniformly bounded finite-coordinate projections imply unconditional convergence of the basis expansions; for real scalars, the sign-change constant is at most $2C$.

A normalized basic sequence generates a spreading model when the norms of all finite scalar combinations converge as the first index tends to infinity, uniformly over increasing choices of the indices. The limiting norm on finitely supported scalar sequences is unchanged by increasing relabellings of coordinates. Completing this norm gives a Banach space with its canonical basic sequence. A diagonal application of the finite-colour infinite Ramsey theorem produces such a model from a subsequence of every normalized basic sequence.

If a bounded subset $K$ of a Banach space is bounded away from zero and zero lies in its weak closure, then $K$ contains a basic sequence. More generally, weak closure may be replaced by closure in any Hausdorff locally convex vector topology weaker than the weak topology.