In the previous post, we were looking for a monster—a nonlinear additive function. We found that such a function is extremely pathological: it is nowhere locally monotone, nowhere continuous and nowhere locally bounded. Worse than that, it's easy to prove that the graph of a monster is dense in
, that is, for every x and y, an arbitrary neighborhood of x contains a point that monster sends arbitrarily close to y.
Recall our attempt to construct a monster. Any additive function is linear on any
Looks familiar? It should be: the definition of B is exactly the definition of a basis of a vector space. Real numbers can be added to each other and multiplied by rationals and, therefore, form a vector space over
After a streak of bad and unlikely properties that a monster has, we now got something positive: a monster exists if and only if
But of course. Any vector space has a basis—this is a general theorem almost immediately following from the Zorn's lemma. The basis we are looking for even got a name of its own: Hamel basis.
At last we stumbled across the whole family on monsters. Specifically, there exists a set
where only finite number of
Take an arbitrary function
that is, f is additive. Intuitively,
is an especially weird monster function: it takes only rational values!
Note that almost all additive functions are, after all, monsters—only very small sub-set of them is linear.
Hey, thanks for these, I was actually wondering how bad would such a function be and you cleared it up for me, nice one!
ReplyDeleteI have no idea how I stumbled onto this, but I'm glad I did. It's wonderful.
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