Approximation property

The approximation property is a property of Banach spaces in which it comes to the approximation of compact operators by linear operators of finite rank. It was forty years an open problem whether all Banach spaces have this property. A closely related problem is the question of whether all separable Banach spaces having a Schauder basis.

How it started

The history of this term begins on November 6, 1936. Banach used in the Scottish Café to Lwów, Lemberg time to ponder math problems. To document these problems, a notebook was purchased, in which not only the mathematical elite Lviv finds, but also problem formulations of John von Neumann, Maurice René Fréchet or Pavel Sergeyevich Alexandrov. Sometimes prices as " two small beer " or " a bottle of wine " were promised to solve the problems. Because of the cafés This book is called the Scottish book and could be saved after the war, (see mass killings in Lviv in the summer of 1941, German occupation of Poland 1939-1945). On November 6, 1936 Stanisław Mazur contributed the following issue number 153 a:

Let be a continuous function for and is. Is there a finite number of integers such that

For everyone?

Stanisław Mazur added that this statement is clear if f has continuous derivatives possess. The price for a solution of the general case was a living goose.

In this formulation, a function of two variables as a sum of products of functions with only one variable is approximated. The problem can therefore be a relationship with tensor suspect. In fact, as Alexander Grothendieck worked in the 50s supernatural topologies on tensor products of locally convex spaces, he found two such topologies and a property that should be locally convex spaces, so these two topologies coincide. For this purpose, it would suffice if every Banach space would have this property. This is the so-called approximation property, which can be defined without recourse to the notion of the tensor product:

Definition of the approximation property

A Banach space E has the approximation property if there is to be any compact set and each a steady, linear operator of finite rank, such that for all.

Equivalent formulation

A Banach space E has exactly then the approximation property if, for every Banach space F and every compact operator and each a steady, linear operator with finite rank.

Limited and metric approximation property

Is it possible to limit the norm of the approximating operators T in the above definition even by a constant, then we say that the Banach space have the limited approximation property. Can you even do this with the constant 1, it is called the metric approximation property.

Banach spaces with Schauder basis

Banach spaces with Schauder basis have the limited approximation property. The converse is not true, as Stanislaw Szarek 1987 could show on the basis of an example.

Thus, most classical Banach spaces have the approximation property:

• Hilbert spaces have the approximation property.
• Is a measure space and so Lp has the approximation property, in particular the sequence spaces have the approximation property.
• The space of all null sequences has the approximation property.
• If a completely regular space, so has the space of bounded, continuous functions with the supremum norm, the approximation property.

Locally convex spaces

The approximation property can be extended to locally convex spaces as follows. A locally convex space has the approximation property if the space of linear operators of finite rank with respect to the topology of uniform convergence on relatively compact subsets of the completion of E is dense in the space of continuous linear operators. That is continuous and linear, a neighborhood of zero and relatively compact in the completion of, so there is a linear operator of finite rank, such that for all.

Permanenzeigenschaften

• Is a family of locally convex spaces with approximation property, as well as the product space have ( with the product topology ) and the direct sum ( with the final topology) the approximation property.
• Have and the approximation property, so also the injective tensor product has the approximation property.
• Are and metrizable locally convex spaces with approximation property, as well as the projective tensor product has the approximation property.
• The completion of a room with approximation property also has the approximation property.
• Have so that and the approximation property Be and Banach spaces. Then also, the space of compact operators, and, the space of trace class operators, the approximation property.

Spaces without approximation properties

Grothendieck observed that the question whether all the Banach spaces have the approximation property, is to issue 153 of the Scottish book equivalent, but they could not be clarified. It was not until twenty years later found out this problem by the Swedish mathematician Per Enflo a negative solution. At the same time, this showed that there must be Banach spaces without Schauder basis. Shortly after the publication of his work traveled by Enflo to Warsaw and took the promised goose.

The example of Per Enflo was "constructed". Meanwhile, one also knows ' prominent ' Banach spaces without approximation properties. In 1981, Andrzej Tomasz Szankowski show that the space of bounded linear operators does not have the approximation property over an infinite-dimensional Hilbert space.

Every Banach space has a closed subspace which does not have the approximation property. The case is here, of course, take out, since this is a Hilbert space.

Swell

• English version of the Scottish book (PDF, 3.1 MB)
• P. Enflo: A counterexample to the approximation property in Banach spaces, Acta Mathematica 130 (1973), 309-317
• A. Szankowski: B ( H) does not have the approximation property, Acta Mathematica 147, 89-108 (1981).
• Robert E. Megginson: An Introduction to Banach Space Theory, Springer 1998
• H. Jarchow: Locally Convex Spaces, Teubner, Stuttgart 1981 ISBN 3-519-02224-9
• S. Szarek: A Banach space without a basis Which Has the bounded approximation property, Acta Math 159 (1987), 81-98

• Functional Analysis