Quick description
When one is trying to find a proof of a mathematical statement, it can be surprisingly helpful to think about the converse of that statement as well. The reason is that an understanding of the converse can give important information about what a proof of the original statement would have to be like, thereby speeding up the search for it.
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Example 1
This may seem a slightly artificial example, but it came up recently in a research problem, and thinking about the converse was an essential step in finding a solution.
Suppose that you have a norm on
and you would like to prove that
for every
belonging to some subset
. Suppose also that you want to do this by estimating the norms of at most
points. (This situation occurred because the norm in question was randomly defined, and it was not possible to ask for too many events to occur simultaneously – at least if one wanted to avoid understanding the very subtle dependencies between those events.) The obvious method is to choose some subset
of
consisting of at most
points, and to run an argument with the following general structure:
every element of
can be approximated (in a suitable sense) by an element of
;
if two elements of
are close (in that same sense) then their norms are close;
the norm of every point in
is smaller than
.
Now let us think about whether this scheme of proof is necessary. That is, if has the property that if every point in
has a small norm then so does every point in
, does it follow that every point in
can be approximated, in some suitable sense, by a point in
?
The answer is an emphatic no: one soon realizes that if the norm of every point in is at most 1, say, then the norm of every point in the convex hull of
is also at most 1. Armed with that observation, we can go back to the original problem with a potentially much more flexible method of proof:
every element of
belongs to the convex hull of
;
every element of
has norm at most 1.
However, if we are sensible, we should learn our lesson and again investigate the converse. Suppose that does not lie inside the convex hull of
. Is it still possible that the norm restricted to
could control the norm restricted to
?
The answer turns out to be no. If some point lies outside the convex hull of
, then the Hahn-Banach separation theorem implies that there is a linear functional
such that
, but
for every
. Thus, the seminorm
is at most 1 everywhere on
but greater than 1 somewhere on
. And provided
is bounded we can easily convert that into a norm with the same property.
What we learn from this is that if we want to find a set and deduce from the fact that the norm of every vector in
is small that the norm of every vector in
is small, then, unless we know, and can use, further information about the norm
(which in our problem we could not), we are forced to use the second method above. Thus, we can stop wasting time searching for alternative approaches.