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## Über dieses Buch

This is an collection of some easily-formulated problems that remain open in the study of the geometry and analysis of Banach spaces. Assuming the reader has a working familiarity with the basic results of Banach space theory, the authors focus on concepts of basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology and nonlinear geometry.
The main purpose of this work is to help in convincing young researchers in Functional Analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems. Inside the Banach space area, the text should help expose young researchers to the depth and breadth of the work that remains, and to provide the perspective necessary to choose a direction for further study.

Some of the problems are longstanding open problems, some are recent, some are more important and some are only local problems. Some would require new ideas, some may be resolved with only a subtle combination of known facts. Regardless of their origin or longevity, each of these problems documents the need for further research in this area.

## Inhaltsverzeichnis

### Chapter 1. Basic Linear Structure

A sequence {e i } i = 1 in a Banach space X is called a Schauder basis for X if for each x ∈ X there is a unique sequence of scalars {α i } i = 1 such that $$x =\sum _{ i=1}^{\infty }\alpha _{i}e_{i}$$. If the convergence of this series is unconditional for all x ∈ X (i.e., any rearrangement of it converges), we say that the Schauder basis is unconditional . This is equivalent to say that under any permutation $$\pi: \mathbb{N} \rightarrow \mathbb{N}$$, the sequence {e π(i)} i = 1 is again a basis of X.
Antonio J. Guirao, Vicente Montesinos, Václav Zizler

### Chapter 2. Basic Linear Geometry

A subset K of a Banach space X is said to be a Chebyshev set if every point in X has a unique nearest point in K. In such a case, the mapping that to x ∈ X associates the point in K at minimum distance is called the metric projection.
Antonio J. Guirao, Vicente Montesinos, Václav Zizler

### Chapter 3. Biorthogonal Systems

In this chapter we review several problems on biorthogonal systems in Banach spaces, i.e., families $$\{x_{\gamma },f_{\gamma }\}_{\gamma \in \Gamma }$$ in X × X , where X is a Banach space, such that 〈x α , f β 〉 = δ α, β whenever α and β belong to $$\Gamma$$. Here, δ α, β  = 1 if α = β and 0 otherwise. Note that Schauder bases, together with their functional coefficients, are examples of biorthogonal systems. The theory of biorthogonal systems is crucial for understanding the structure of Banach spaces, in particular of nonseparable ones. Many problems in this area are widely open. In the nonseparable case the theory of biorthogonal systems often goes as deep as to the roots of Mathematics, i.e., they use special axioms of Set Theory. In this respect we refer, for the most basic information, to, e.g., [HMVZ08, pp. 148 and 152] or [To06].
Antonio J. Guirao, Vicente Montesinos, Václav Zizler

### Chapter 4. Differentiability and Structure, Renormings

In this chapter we review some problems on smoothness, rotundity, and its connection to the structure of spaces. We recommend, for example, [BenLin00, DeGoZi93, Fa97, FHHMZ11, HMVZ08], and the recent book [HaJo14] for this area.
Antonio J. Guirao, Vicente Montesinos, Václav Zizler

### Chapter 5. Nonlinear Geometry

In this chapter we review several problems in the area of nonlinear structure of Banach spaces.
Antonio J. Guirao, Vicente Montesinos, Václav Zizler

### Chapter 6. Some More Nonseparable Problems

The following is the definition of a type of Schauder basis that works also for nonseparable spaces. It is due to P. Enflo and H. P. Rosenthal in [EnRo73].
Antonio J. Guirao, Vicente Montesinos, Václav Zizler

### Chapter 7. Some Applications

Schauder’s theorem asserts (see the comments to Problem 274 above) that if C is a compact convex set in a Banach space X and if f is a continuous mapping from C into C, then f has a fixed point , i.e., there is x ∈ C such that f(x) = x (cf., e.g., [FHHMZ11, p. 542]).
Antonio J. Guirao, Vicente Montesinos, Václav Zizler

### Backmatter

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