Month: April 2026

04/19/26

How do you intuitively explain the fact that the bidual of an infinite dimensional vector space is bigger than the vector space itself?

My answer to a question on Quora:
“How do you intuitively explain the fact that the bidual of an infinite dimensional vector space is bigger than the vector space itself?”.
MY ANSWER:
It is a very deep question indeed. I have to admit that I cannot give any intuitive explanation which is simpler than a reduction to some basic set theory.
To make the question as close to the set theory as possible, let us restrict our attention to the case of a vector space V over the field F_2 of two elements. If my memory does not betray me, it is an old result by Paul Eklof that existence of a basis in an arbitrary vector space over F_2 is equivalent to the Axiom of Choice (it is easy in one direction: the Axiom of Choice, in the form of the Zorn Lemma, implies the existence of a basis). So, let us accept it, and let B be a basis in V, which means that every vector v in V is defined by its support in B, that is, by the (finite) set of elements in B which sum up to v. Therefore V has the same cardinality as the set of finite subsets of B; if B is infinite, than it is easy to prove that the set of all finite subsets of B has the same cardinality as B, hence V has the same cardinality as B.
Now let us look at the dual space V*, that is, the set of all linear functionals from V to F_2. Each such functional is uniquely determined by its support in B, that is, by the set of basis vectors where it takes value 1. Therefore V* is in one-to-one correspondence with the set 2^B of all subsets in B. But it is a classical result by Cantor that 2^B has larger cardinality than B and hence V* has larger cardinality than V. Of course, the cardinality of the bidual V** is even larger.
The case of an arbitrary field F can be handled in a similar way, but we will need to deal with the cardinality of the set F^B.
Is this intuitive? Well, it is intuitive for me because it was the first thought that crossed my mind. But I am not a set theorists and I have no idea whether the same can be proven without the Axiom of Choice. Also, the term “infinite dimensional vector space is ambigous: does it mean “a vector space without a finite basis” or “a vector space with an infinite basis”? And can anything that intimately involves the Axiom of Choice be called intuitive?
COMMENTS: Someone commented on that: “Without Axiom of Choice, a vector space can have trivial dual (containing only the null functional).” If true (I am not a set theorist and cannot judge), this even more emphasises the crucial role of Axiom of Choice. Also, I had perhaps mention in my answer that the situation could be very different if we consider topological vector spaces and continuous linear functionals on them.
04/18/26

Why do people have to learn algebra? You have no use for it.

My old post on Quora, an answer to question

Why do people have to learn algebra? You have no use for it.

Now I would perhaps slightly modify it, see my Addition at the end of the present post.

It could be argued indeed that it is wrong that everyone is forced to learn algebra at school.

However, without (pretty basic, between us) school algebra further study mathematics, or statistics, or computer programming is impossible.

Not learning algebra dramatically narrows further educational choices. Catching up later is very difficult — if you have not learnt algebra as a child or, at the latest, in your teen years, to do this later is of course possible but requires a degree of determination and self-discipline not normally found in general population.

Therefore, without algebra, the education system is less democratic. This is already a heavy price to pay.

In general, mathematical education is increasingly the issue of democracy.

Proper mastery of mathematics gives a person the ability to create in the head mental images of complex systems and operate with them, see them. We are surrounded by complex systems, we drown in them — in stuff starting from smartphones to the Internet and social media . Very complex computer programs make more and more decisions on peoples’ lives – whether someone can be hired to a particular job, or given a bank credit, or sold a a particular insurance policy, and so on.

Properly mathematically educated people are sighted among the blind.

If you wish to stay blind – it is your choice. Or you may think that it is entirely your choice. Actually it could happen that it is already pre-determined by a dismally bad education which, most likely, had been offered to you so far — otherwise, perhaps, you would not ask the question “ Why do people have to learn algebra?”.

Added 18.04.26:

I think the question needs a reformulation and should be

Why at least some people have to learn algebra?

with the  answer

Because  the human race still has to use it.

04/9/26

How surprised would you be if mathematicians discovered a 27th sporadic finite simple group?

My answer to a question on Quora:

How surprised would you be if mathematicians discovered a 27th sporadic finite simple group?

I would be really surprised.

I am one of the few people in the world who had reason to read, and spent some time reading, certain parts of the proof of the Classification of Finite Simple Groups (CFSG), and I have some basic understanding of what is going on. A few points:

  1. Many parts of the proof of the CFSG (in its various versions) are done by induction, by considering a minimal counterexample: a smallest, by order, finite simple group which is not on the list. The arguments involved are fine tuned at identification of a new finite simple group, if one exists. The fact that this has not happened in the last 30 year is quite reassuring.
  2. Some “classical” groups are more sporadic (in the sense that their properties are quite abnormal) than most sporadic group (a good example is the projective special linear group  which should be seen as Mathieu group ). In that sense there are already more than 26 sporadic simple groups.
  3. I heard some good mathematicians suggesting that at least some of the sporadic groups are likely to belong to infinite series of a new kind of algebraic structures still unknown to us, but some of which have happened, by chance, be groups — the same way as the alternating group has happened to be the linear group , living simultaneously in two different universes.

The third point, if confirmed, would be really exciting and likely to have long lasting impact on mathematics.