Approximation property

The approximation property is a property of Banach spaces that is about the approximation of compact operators by linear operators of finite rank . For forty years it was an open problem whether all Banach spaces had this property. A closely related problem is the question of whether all separable Banach spaces have a shudder basis .

How it started

The history of this term begins on November 6, 1936. Stefan Banach used to ponder mathematical problems in the Scottish Café in Lwów , then Lemberg. To document these problems, a notebook was purchased that not only includes Lviv's mathematical elite , but also problem formulations by John von Neumann , Maurice René Fréchet and Pawel Sergejewitsch Alexandrow . To solve the problems, prizes like “two small beers” or “a bottle of wine” were sometimes offered. This book is called the Scottish Book because of the café and was saved after the war (see also mass murders in Lemberg in summer 1941 , German occupation of Poland 1939–1945 ). On November 6, 1936, Stanisław Mazur entered the following problem number 153:

Let be a continuous function for and be . There are finitely many numbers such that ${\displaystyle f(x,y)}$ ${\displaystyle 0\leq x,y\leq 1}$ ${\displaystyle \epsilon >0}$ ${\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{n},c_{1}\ldots ,c_{n}}$

${\displaystyle {\big |}f(x,y)-\sum _{k=1}^{n}c_{k}\cdot f(a_{k},y)f(x,b_{k}){\big |}\leq \epsilon }$ for everyone  ? ${\displaystyle 0\leq x,y\leq 1}$

Stanisław Mazur added that this statement is clear if f has continuous derivatives. The price of a common case solution was a live goose.

In this formulation, a function of two variables is approximated as the sum of products of functions with only one variable. The problem therefore suggests a relationship to tensor products . Indeed, when Alexander Grothendieck was working on natural topologies on tensor products of locally convex spaces in the 1950s , he found two such topologies and a property that the locally convex spaces should have so that these two topologies coincide. For this it would be sufficient if every Banach space had this property. This is the so-called approximation property, which can also be defined without resorting to the term tensor product:

Definition of the approximation property

A Banach space E has the approximation property if there is a continuous , linear operator of finite rank for each compact set and each , so that for all . ${\displaystyle K\subset E}$ ${\displaystyle \epsilon >0}$ ${\displaystyle T:E\rightarrow E}$ ${\displaystyle \|Tx-x\|\leq \epsilon }$ ${\displaystyle x\in K}$

Equivalent formulation

A Banach space E has the approximation property if and only if for every Banach space F and every compact operator and each a steady , linear operator finite rank with there. ${\displaystyle S:F\rightarrow E}$ ${\displaystyle \epsilon >0}$ ${\displaystyle T:F\rightarrow E}$ ${\displaystyle \|T-S\|\leq \epsilon }$

Constrained and metric approximation property

If one can even restrict the norm of the approximating operators T in the above definition by a constant, then one says that the Banach space has the restricted approximation property . If this can even be achieved with the constant 1, one speaks of the metric approximation property .

Banach rooms with a shudder base

Banach spaces with a shudder basis have the restricted approximation property. The reverse is not true, as Stanislaw Szarek was able to show in 1987 using a counterexample.

Most classical Banach spaces thus have the approximation property:

• Hilbert spaces have the approximation property.
• If a measure space is and , then L ^{p has} the approximation property, in particular the sequence spaces have the approximation property. ${\displaystyle (X,\mu )}$ ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle (X,\mu )}$ ${\displaystyle \ell ^{p}}$
• The space of all zero sequences has the approximation property. ${\displaystyle c_{0}}$
• If a completely regular space is , then the space of bounded, continuous functions with the supremum norm has the approximation property. ${\displaystyle X}$ ${\displaystyle (C^{b}(X),\|\cdot \|_{\infty })}$ ${\displaystyle X\rightarrow {\mathbb {K} }}$

Local 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 the linear operators of finite rank with respect to the topology of the uniform convergence on relatively compact sets of the completion of E is dense in the space of the continuous linear operators. Ie is continuous and linear, a neighborhood of zero and relatively compact in the completion of , so there is a linear operator of finite rank , so that for all . ${\displaystyle E}$ ${\displaystyle S:E\rightarrow E}$ ${\displaystyle U\subset E}$ ${\displaystyle K\subset E}$ ${\displaystyle E}$ ${\displaystyle T}$ ${\displaystyle E}$ ${\displaystyle T(x)-S(x)\subset U}$ ${\displaystyle x\in K}$

Permanent properties

• If a family of locally convex spaces with approximation property, then the product space (with the product topology ) and the direct sum (with the final topology ) also have the approximation property. ${\displaystyle (E_{j})_{j}}$ ${\displaystyle \Pi _{j}E_{j}}$ ${\displaystyle \textstyle \bigoplus _{j}E_{j}}$
• If and have the approximation property, then the injective tensor product also has the approximation property. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\otimes _{\epsilon }F}$
• If and are metrizable locally convex spaces with approximation properties, then the projective tensor product also has approximation properties. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\otimes _{\pi }F}$
• The completion of a space with approximation property also has the approximation property.
• Let be and Banach spaces such that and have the approximation property. Then , the space of compact operators , and , the space of trace class operators also have the approximation property. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\,'}$ ${\displaystyle F}$ ${\displaystyle K(E,F)}$ ${\displaystyle E\to F}$ ${\displaystyle (N(E,F).\|\cdot \|_{1})}$ ${\displaystyle E\to F}$

Per Enflo accepts the award.

Spaces without approximation property

Grothendieck noted that the question of whether all Banach spaces have the approximation property is equivalent to Problem 153 in the Scottish Book, but could not answer it. It was not until twenty years later that the Swedish mathematician Per Enflo found a negative solution to this problem . At the same time this showed that there must be Banach spaces without a shudder base. Shortly after his work was published, Per Enflo traveled to Warsaw and received the promised goose.

Per Enflo's example was 'constructed'. Meanwhile, we also know 'prominent' Banach spaces without approximation property. In 1981 Andrzej Tomasz Szankowski was able to show that the space of bounded linear operators over an infinite-dimensional Hilbert space does not have the approximation property.

Every Banach space has a closed subspace that does not have the approximation property. The case is of course to be removed here, since it is a Hilbert space. ${\displaystyle \ell ^{p},p\in [1,\infty ]\setminus \{2\}}$ ${\displaystyle p=2}$

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