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\footline={\hss\tenrm\folio\hss}
{\rightline {\eightrm In reality,
            all this has begun because of A.Grothendieck.}}

        \topmatter
  %      \centerline{\bf \
  \heading{\bf ON BASES AND APPROXIMATION CONDITIONS IN BANACH SPACES\linebreak
  %}  %\endhead
%\centerline{ \eightpoint \bf
(trying giving a short survey${ }^{*)}$)}\endheading
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          \author { Oleg I.Reinov${{ }^\dag}$}  \endauthor
\address\newline
Oleg I. Reinov \newline
Department of Mathematics\newline
St Petersburg University\newline
St Peterhof, Bibliotech pl 2\newline
198904  St Petersburg, Russia
\endaddress

\email
orein\@orein.usr.pu.ru
\endemail

\thanks
                          %\tenrm
${ }^{*)}$%
This small lecture was given in some places of Sweden,
and was written here just before (or even during)
those talks. So, please, the reader (if any...)
be not very strict to the author because of not every place of the text is
looking nice enough, or maybe even is "enough not nice". Especially,
maybe, this concerns my dreadful English (but not only?..).
\endthanks
\thanks
I would like to bring my sincere acknowledgements to the Royal Institute
of Technology, and especially to Professor Ari Laptev, for providing me
the excellent working conditions during my stay in Stockholm in 2000.
Besides, I wish to say many thanks to Professor Sten Kaijser and to
Professor  Lars Hedberg. They were so kind to me that have not been
afraid of my poor English and given me a possibility to visit their
departments. These visits were certainly very useful for me.
And also I am very grateful to my Stockholm landlady
Teresia Sommer for her hospitality.
\endthanks

\thanks
               %\tenrm
${{ }^\dag}$ The work was done with partial support of the Ministry of the
general and professional education of Russia (Grant 97-0-1.7-36).
      \endthanks

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\footnote""{${ }^\ddag$
AMS Subject Classification:  47B10. Hilbert--Schmidt operators,
trace class operators, nuclear operators, p-summing operators, etc.
}
\footnote""{${ }$
Key words: bases, approximation properties.
}


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One or two times we shall go out of the main theme of the talk
to formulate very interesting results, directly (for the first sight)
not connected with the objects, mentioned in the title.
They are so interesteing that, by the way, it will be hard enough
to leave them away.
On the other hand they, in any case, are closely related to those
approximation conditions.

Let us begin with recalling that {\it a basis} in a Banach space $X$
is a sequence $(x_n)_1^\infty$ with the property that any $x\in X$ has
an unique representation of the form $x=\sum_{n=1}^\infty a_n x_n$ in $X.$
The functionals $x_n': x\mapsto a_n$ are called {\it the coordinate
functionals}\ of the basis $(x_n).$
It is not evident (and this is a result of Banach), that these functionals
are continuous; hence the so called operators of partial sums
$P_N:x\mapsto \sum_{n=1}^N a_n x_n$ are continuous too and, therefore, they
are continuous linear projections from $X$ onto the linear (closed) span
of $(x_n)_1^N,$\ $N=1,2,\dots.$ Of course, any space with a basis is
separable.

The notion         % NB:"there is a notion or many notions"
of the basis was firstly introduced by Schauder in 1928, when he proved
that in the space $C[0,1]$ of all continuous functions with the natural
$sup$-norm the "integrated" Haar system is a basis in the above sense
(Haar, himself, showed in his thesis that his system is "almost basis"
in this space: he proved that every continuous function uniquely expands
into an unifomly convergent series with respect to the Haar system;
but Haar functions are not continuous! -- so they do not give a basis for the space
$C[0,1]).$ An year later, in 1929, Schauder showed that the Haar system,
itself, is a basis in each space $L^p[0,1]$\footnote{
These bases in $L^p$ (when $1<p<+\infty$)
are even unconditional, i.e. they remain being bases under any
permutation, but we are not going to discuss this theme; see any
corresponding literature.},
\, $p\in [1,+\infty)$
(let us recall another, rather hard in proving, examples:
the trigonometrical system is a basis in every space
$L^p[0,1],$\, $p\in (1,+\infty),$ but not in $C[0,1]$ or $L^1[0,1]).$
Now it is known that all the classical Banach spaces have bases.
For example, a basis in the disc-algebra $A$ (the space of all analytical in
open unit disk and
continuous in the closed disk functions) was constructed by Bochkarjov;
in $H^1$ -- the Hardy space -- by B.Maurey. In the spaces $H^p,$
for the other exponents $p\in (1,\infty),$ bases were known from early days of the birth of the
notion of the basis.
%But do not think that .............

The question of the existence of a basis in every Banach space was posed by
Banach in 1932 (his famous work!). Only in 1972 it has become real to look
at an example of the space without basis (P. Enflo), and moreover, ---
without the so called metric approximation property (and even without the
approximation property; see below).

%$\al$ WRITE ON .....

Those operators of partial sums $(P_N)$ possess the properties

1) $P_N \to \id_X$ pointwise;

2) $\sup_N ||P_N||<+\infty;$

3) $(P_N)$ converges to $\id_X$ uniformly on every compact set $K$
in $X;$

4) $\dim P_N(X)=N$ and $P_N(X)\sbs P_{N+1}(X).$

The property 1) is evident; 2) is a corrolary of Banach--Steinhaus theorem
and 1);
it follows from 1), taking in account 2), the condition 3); 4) is clear.

Just for a couple of minutes, let us forget that the cause of appearing
the assertions 1)--3) was the basis, and consider any sequence $(P_N)$
of finite rank
operators in $X,$ satisfying the condition 1) (consequently, 2) and 3)).
A (separable) Banach space $X$ in which the condition 1) (or 3))
is fulfilled is said to be {\it the space with
the bounded approximation property,}\,
BAP in short. Cleary, the existence of a basis in a Banach space
implies that this space has the BAP. On the other hand, if $(P_N)$ is
a sequence of projections which satisfies 1)--4), then it is easy to
construct the corresponding basis, for which $(P_N)$ is the sequence of
operators of partial sums (just take, for each $N=1,2,\dots\,,$ \, nonzero
$x_N\in P_N(X)\setminus P_{N-1}(X),$ where $P_0=0).$
But in general case, when $(P_N)$ is a sequence of finite rank operators
satisfying only 1),
it becomes a problem to construct a basis in the space $X.$

Long ago, A.Pelczy\`nski showed that {\it any Banach space with a basis
is a complemented subspace of a $($universal and isomorphically unique$)$
space with the $($universal$)$ basis.} Besides, he proved
{\it that every separable Banach space
with the BAP can be imbedded, complementably, in a space with a basis.}
But it was, and was long enough, not known whether a complemented
subspace of a space with a basis has a basis itself.
Finally, Szarek was successed in showing that the answer to the last question
is "no", so solving this (and some other problems) in negative.

Of course, any finite dimensional space has a basis, and not one.
But, in any case, there are some natural notions and questions,
related to them, in finite dimensional theory
too. One can define a quantitative notion connected with a basis
(in any Banach space, not only in finite dimensional one).
This can be done as follows.

Let $X$ be a Banach space, and $(x_n)$ is its basis.
Then the natural projections (operators of partial sums),
$P_N: x\mapsto \sum_1^N <x'_n,x>x_n,$ as was already said,
are bounded linear operators in $X.$
{\it The basic constant of the basis} \, $(x_n)$ is defined as the
supremum of the norms $||P_N||$ over all $N;$
it is denoted by $bc\,(x_n).$
If $X$ has a basis then {\it the basic constant of the space}
$X$ is not anything as the number $bc\,(X):= \inf \, bc\,(x_n),$
where {\it inf}\ is taken over all bases in the space $X.$
It is very interesting to consider this number in the case
of a finite dimensional space. It is rather not hard to show that for any
$n$-dimensional normed space $X$ we always have $bc\, (X)\le \sqrt n$
(the reason is that any such space is close enough
to the Euclidean space $l^n_2,$ namely, is $\sqrt n$-close in the sense
of "Banach--Mazur distance", but we are not going to discuss what the last
phrase means, because that would lead us far enough from the main theme
of our considerations, although to a very nice theory;
--- it is not our aim).

So, a very old question was (in connection with the last inequality)
a question of the assymptotical precision of that estimate.
And, by using probabilistic methods, Gluskin and Szarek showed that
indeed, there is a $C>0$ so that for some sequence $(X_n)_{n=1}^\infty$
of finite dimensional spaces with $\dim X_n=n$\
one has $bc\,(X_n)\ge C\sqrt n.$
By the way, they proved that, in a sense, "no subspase of $X_n$
of large enough dimension is complemented in $X_n$"
(that means, in particular, that any subspace of $X_n$
of the half dimension has the property that there is no projection
from $X_n$ onto it with norm less then $c_0\sqrt n,$
for some absolute constant $c_0.)$

Well, the last mentioned finite--dimensional result leads to a very
old classical question if such a situation is possible in the infinite
dimensional spaces. We will give here only two striking results
in this direction.

Firstly, G.Pisier, answering one of the question of A.Grothendieck,
showed that there exists an infinite dimensional Banach space $G$
all of whose finite dimensional subspaces lie in $G$ very bad:
there is a constant $c>0$ such that for any subspace $F$ in $G$ the best
projection from $G$ onto $F$ has the norm more than $c\sqrt{\dim F}$\,(! --
it is well known that every $n$-dimensional subspace of any Banach space
is $\sqrt n$-complemented).

Secondly, B.Maurey with a coauther showed that there exists an infinite
dimensional Banach space $B$ without any complemented subspaces; more
precisely, if only a subspace of $B$ is not finite dimensional itself,
or is not finite codimensional, there is no linear continuous
projection from $B$ onto this subspace (!!).

But now, let us return from the sky filled with striking results
to our bases and approximation conditions...

Not knowing whether every separable Banach space has a basis,
in 1955 A.Grothendieck has introduced more general notions, ---
AP (approximation property) and BAP (bounded approximation property), ---
to go around the basis problem and
to cover the situation when the spaces under
consideration are not separable
(and also, in particular, to prove many facts which it was hardly
to get without those additional restrictions; well, when it was possible
to prove
a result with only the AP, he used it, but in many cases it was needed
a stronger version of
the approximation in Banach spaces, so he used the MAP).
Here are the definitions
(in a sense, inspiring by the above discussion on bases and BAP in
separable spaces, i.e. by the condition 1) and, to be exact,
by conditions 2) and 3): 3) will give us the AP, while 3) with
something like 2) will give the BAP and the MAP).
So:

They say that
a Banach space $X$ has {\it the approximation property}\ (AP), if

$(*)$ for each $\e>0$ and every compact set $K\sbs X$
there exists a finite rank operator $P$ in $X$ such that
for each $x\in K$ one has $||Px-x||<\e.$

The space $X$ has the BAP ({\it bounded approximation property}),
if there is a constant $C>0$ such that $(*)$ is fulfilled with
the additional condition $||P||\le C;$ if we can take $C=1,$
then we say that the space $X$ has {\it the metric approximation
property}\ (MAP).

P.Enflo (we said about this a little before) proved that
{\it there exists a separable reflexive space without
the}\ AP, using, in particular, the following nice result of
A.Grothendieck
(certainly, this Grothendieck's theorem is not the main
tool in the difficult proof of Enflo):
{\it for reflexive Banach spaces the}\ AP {\it is just the same as the}\
MAP.

This last result and some other assertions of A.Grothendieck led
him to the following assumption:
{\it if a weakly compact operator, acting in some Banach spaces,
can be approximated, uniformly on each compact, by finite rank operators,
then it also can
be approximated, uniformly on each compact, by finite rank operators,
whose norms are bounded by a constant, depended only on the given
weakly compact operator.}\ I was successed in giving a counterexample
to this assumption, constructed a separable space and a compact operator
in this space, which is "approximable", but not
"bounded approximable".
By the way, this space has the AP, but has not the BAP.

As the matter of fact, {\it the first example of the space with the}\
AP {\it and
without}\ BAP {\it was found by T.Figiel and W.B.Johnson,}\ who also answered
another, close to the problem, question of A.Grothendieck (related
to the so called nuclearity of operators in Banach spaces).
The last facts led to many interesting questions in the geometrical theory
of operators in Banach spaces. But they are not in the sphere of our
considerations here. We would like to mention here only the following
result which generalises the Figiel-Johnson theorem.

Let $I(X,X)$ be a closed subspace of the space $L(X,X)$ of all linear
bounded operators in a separable space $X.$
We now know that the identity map $\id_X$ is not nessesary can be
compactly approximated by a bounded sequence of finite rank operators,
even if $X$ has the AP. But maybe such a situation is impossible if instead
of finite rank operarors we will consider the operators from the
space $I(X,X)$ (if this space $I$ is large enough, but not the whole
$L(X,X)$)? However, in general, the situation here is just the same as in the
Figiel-Johnson theorem, e.g.:
{\it there exists a separable Banach space with the}\ AP {\it such that
the identity map of this space can not be approximated, in the topology
of compact convergence, by weakly compact operators.}\
With the formulation of this result (its proof is somewhat different
from the one of Figiel-Johnson) I am going to finish the discussion
around bases, AP, BAP etc.. And to end the talk, I would like to propose two,
more that twenty years old, open questions.

1) Let $X$ be a separable Banach space. Is it true that if the dual
$X^*$ has the AP, then this dual has the BAP?

2) Let $T$ be a linear bounded operator from a Banach space $X$
to a Banach space $Y$ such that there exists a continuous
factorization of $T^{**}$ of the kind
$$ X^{**}\to l_p \to Y^{**},
$$
where $1<p<+\infty,$\,  $p\neq2.$
Does there exist a similar factorization
for the operator $T: X\to Y?$

Note, finally, that if the space $X^*$ is separable, then the answer to
the question 1) is positive (A.Grothendieck).
As to the second question, the answer is negative for the cases
$p=1$ or $p=\infty.$ Trivially, the answer is positive, when $p=2.$

The second question does, as we could think, not deal with the AP's;
but it is only a first sight on it (deeply inside it the AP seats,
as well as some different generalizations of the approximation property
do)...

\nopagebreak        \medskip
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    \centerline{REFERENCE}    \eightpoint
    %\bigpagebreak
     \smallskip

\nopagebreak
\ref \no \by Grothendieck A. \pages 196+140
\paper  Produits tensoriels topologiques et espaces nucl\'eaires
\yr 1955\vol  16
\jour  Mem. Amer. Math. Soc.
\endref
\nopagebreak

\enddocument

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