# Download Introduction to the Theory of Bases by Jürg T. Marti PDF

By Jürg T. Marti

Since the book of Banach's treatise at the thought of linear operators, the literature at the idea of bases in topological vector areas has grown vastly. a lot of this literature has for its beginning a question raised in Banach's ebook, the query no matter if each sepa rable Banach house possesses a foundation or now not. The thought of a foundation hired here's a generalization of that of a Hamel foundation for a finite dimensional vector area. For a vector house X of limitless size, the concept that of a foundation is heavily with regards to the convergence of the sequence which uniquely correspond to every element of X. hence there are kinds of bases for X, in accordance with the topology imposed on X and the selected kind of convergence for the sequence. even if nearly 4 a long time have elapsed seeing that Banach's question, the conjectured life of a foundation for each separable Banach house isn't but proved. however, no counter examples were stumbled on to teach the life of a distinct Banach area having no foundation. although, as a result of obvious overconfidence of a gaggle of mathematicians, who it's assumed attempted to resolve the matter, we now have many dependent works which convey the tight connection among the speculation of bases and constitution of linear spaces.

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**Example text**

I CX I 2 vn I max Mt <00, n I i~m Ilcxdl ~ k=l i~m n k =1 ± iEamnq~ n = k =1 - IECim(\ak n 0 IEGmnak k- 1 i n ~ k= 1 ± and the proof is complete. In order to prove the Dvoretzky-Rogers theorem which states that in every infinite dimensional Banach space there exists an unconditionally convergent series which is not absolutely convergent, we prove a geometrical lemma about symmetric convex bodies in the Euclidean space [Rn (by a body we mean the closure of a bounded open set in [Rn). For the rest of this chapter we assume X to be infinite dimensional.

6. T-Bases 43 Proof. X has a Hamel basis, since each linearly independent set of n elements {x 1, ... , x n } in X forms such a basis. The existence of a biorthogonal sequence of coefficient functionals {xi, ... 17) there is a topological isomorphism T of X onto En. Let {yJ be an orthonormal basis for En. Then X=T-1[t1(TX,Y;)Yi]=it/TX'Yi)T-1Yi' Since the set {T- 1yJ is linearly independent in X, the set {(Tx,y;)} is unique for each x in X. Moreover x{:X--dJ, defined by X{(x)=(Tx,y;) is linear and, by Ix{(x)1 ~ IITlllly;llllxll, bounded.

The existence of a biorthogonal sequence of coefficient functionals {xi, ... 17) there is a topological isomorphism T of X onto En. Let {yJ be an orthonormal basis for En. Then X=T-1[t1(TX,Y;)Yi]=it/TX'Yi)T-1Yi' Since the set {T- 1yJ is linearly independent in X, the set {(Tx,y;)} is unique for each x in X. Moreover x{:X--dJ, defined by X{(x)=(Tx,y;) is linear and, by Ix{(x)1 ~ IITlllly;llllxll, bounded. This implies that {T- 1Yi,Xi} is a basis for X and we are done. Theorem 5. There exists a uniform basis for X if and only if X is finite dimensional.