Re: Is the theory of topological vector spaces still alive?

Tim, I do not agree with you.

1. ...Pick a random scholar in a random academic subject from a random period
in history, and that scholar will assure you that the problem is particularly
bad in that subject and at that period in time.

Of course, what we discuss happens very often in different parts of
mathematics. But this does not mean that everywhere the situation is
the same.
For instance, Stephen mentioned here Banach spaces. Although, I am not
impressed in what happens in the theory of Banach spaces, I
nevertheless
presume to prove that situation there is better than situation in
topological
vector spaces.

My proof is as follows. From the point of view of category theory the
difference between Banach and non-Banach situations is that in the
first case
the theory suggests a convenient class of objects that form a monoidal
closed
category (namely, class of Banach spaces) and for each monoid in this
category
(in this theory monoids are nothing more nor less than Banach
algebras) the
corresponding modules over this monoid form enriched category over the
initial
category (of Banach spaces).

In the theory of topological vector spaces the situation is absolutely
different. For its lifetime this science did not create any class of
spaces,
convenient from the point of view of the customary algebraic
intuition, i.e., a
class that, like the class of Banach spaces, one could put into the
place of
the usual vector spaces in pure algebra.

For those readers who are far from the category theory, this idea
becomes clear
after consideration the construction of algebra of endomorphisms. As
is known,
in pure algebra every module $X$ over an algebra $A$ generates an
algebra
$End_A(X)$ of endomorphisms of $X$ over $A$. This elementary fact
ceases to be
true in topological algebra, if we require algebras and modules to be
complete
(in some sense, general for all algebras and modules), and to have
continuous
multiplication (again in some sense, general for all these objects).
This can
be conveniently illustrated by the following

Exercise. Give a definition of topological algebra and topological
module such
that the following conditions hold:

1) all the topological modules are topological vector spaces and
satisfy some
standard condition of completeness (we need this to provide the
convergence of
natural nets and series);

2) the multiplication operations are continuous in some reasonable
sense;

3) there is a natural procedure that endows the ring $End_A (X)$ of
all
endomorphisms of a given topological module $X$ over a given
topological
algebra $A$ with the structure of topological algebra with respect to
your
definition.

You may be surprised, but up to the last time the only known solution
of this
Exercise in the frame of the theory of TVS was the class of Banach
algebras and
modules. (We can note here that the appearance of Banach spaces is not
a merit
of the theory of TVS, since historically Banach spaces were studied
before the
general topological vector spaces. Moreover it was the narrowness of
the class
of Banach spaces that has lead to the appearance of the theory of
TVS.)


On hearing this one can ask: "What were you, specialists in
topological vector
spaces, doing all this time?" I asked them similar questions, and if
we
translate what they usually answer to the normal langauge, the
translation will
be as follows: "Our counterexamples are more interesting for us. This
is what
we are proud of!"

So my first counter-argument is that the situation in different parts
of
mathematics is not the same, and we can compare it (here I agree with
Stephen).
And in this comparison the situation in the theory of TVS looks
scandalous.

2. It's not clear to me that it's a "serious problem." Rather, it seems to
me to be mostly harmless, a modest price to pay for the numerous good papers
that get written. Moreover, I've suggested more than once already that
"dormant" may be a better word than "dead."

I do not agree with this as well. First, I do not see "numerous good
papers
that get written" in the theory of TVS. And, second, I had not
opportunity to
make an experiment, but I am sure, if I replaced "dead" with "dormant"
in my
paper, the reaction would be the same: irritation.

My counter-argument here is as follows. If the idea that the
hyper-specialization is a modest price for the progress becomes
dominant in
scientific society, and people imply from this that we should not be
too
exacting to what those "hyper-specialists" do, then we inevitably come
to a
situation when those "hyper-specialists" abuse their power.

In practice this abuse looks as follows. When you are a student they
tell you
that this or that mathematical result "is very important and elegant",
and
despite your doubts, you have to spend your time on studying
innumerable
counterexamples (which of course are "the most convincing evidences of
this
beauty"). As a corollary, by the time when you defend your PhD, you
loose your
human nature: you become a robot, that cannot differ useful things
from useless
things, beauty from deformity, and decency from dishonourableness.
Because
questions like "why is your science useful" -- you treat as an
invitation to
bewilder the interlocutor by your professional skill.

My reproach to mathematical society is that there is no culture here
in such
discussions. Mathematicians do not acknowledge their duty to explain
simply and
clearly why their field of interests is useful. Gradually they turn
into
sportsmen whose aim is to impress the audience by their skill, and
nothing is
important besides the skill. Those counterexamples by Enflo and others
are
indeed quite sophisticated. But if we treat them as progress in
science, like
people in TVS do, this becomes a speculation. Because in fact
counterexamples
are evidences of failure.

That is my point.

If you live in the West this problem, I suppose, is not of current
importance
for you, because you may have a lot of possibilities to change your
company and
to find like-minded persons, but when you live in a country like
Russia, you
become completely dependent on the opinion of those "hyper-
specialists", or
perhaps we should say, "skilful swindlers"? :)

So still I am curious if there are any specialist in the theory of
topological
vector spaces, who could explain these oddities in their science?

.

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