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



Tim, it is clear for me that if we take your definitions, then we
never come to conclusion that any field of science is dead. The very
first your conjecture --

Conjecture 1. The study of topological vector spaces was dead during
the period 1970 to the present.
-- is formally not true, since there were some results in this science
after 1970, for example the famous Enflo and Szankowski
counterexamples (1973 and 1981 respectively).

As to the second one,

Conjecture 2. The study of topological vector spaces will continue to
be dead for all time in the future.

-- it is so severe that becomes useless, since we cannot be sure that
anything, or even anyone, will be dead for all time in future.
Even when a person dies we cannot be sure that this will last forever,
since in future the human can contrive something like time machine,
and this person will be alive again. (To say nothing about people’s
religious belief that the death lasts only to the Last Judgement.)

When writing that the theory of topological vector spaces is dead I
had in mind that the current investigations in this science seem to
be useless for the other parts of mathematics. There are many examples
of such situations, for instance Set Theory. Every mathematician now
uses the results of Set Theory. But can we say that Set Theory is
alive? People, who conceptualized this filed of mathematics, have
built some useful construction named now Set Theory, and we can say
that this work is finished by now, since for a long time there were no
useful improvements in this area. At least I did not heard about them
(of course, if you are a specialist in Logic, you can correct me).

And a similar situation is in the theory of topological vector spaces,
but the difference is that, first, the circle of mathematicians who
use their results is much more narrow, and, second, the circle of
those who claim that they “continue investigations in TVS” is a little
bit wider.

So I have been meaning to ask people, in which situations one can say
that this or that theory is dead? The specialists in the theory of
topological vector spaces seem to feel hurt, when I try to speak with
them about the “philosophy” of their science, but how do they expect
people to discuss such questions? -- This is a puzzle for me.

.

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