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

Dear moderators, I again ask you to replace my previous post by this
new one, because I again found mistakes there. Excuse me my English,
and thank you in advance, Sergei Akbarov.

------------------------------------------------------------


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 here 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 by what happens in the theory of Banach
spaces, I nevertheless presume to prove that situation there is better
than 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 (here 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 bjects).
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. (Note by the way 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 may 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. This is not very important, but,
first, I want to say that 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 main 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, who cannot distinguish useful things from useless things,
beauty from deformity, and decency from dishonesty. 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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