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What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$?

I want to say that it is at least $2^{\aleph_0}$, but I have no idea how to sharply pin it down otherwise.

Asaf Karagila
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joshphysics
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1 Answers1

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HINT: The cardinality of $\Bbb{R^R}$ (all the functions from $\Bbb R$ to itself, not just the continuous ones or whatnot) is $2^{2^{\aleph_0}}$. And note that if $V$ is a vector space over an infinite field, with basis $B$, then $|V|=\max\{|B|,|F|\}$.

Asaf Karagila
  • 405,794
  • Wow ok $2^{2^{\aleph_0}}$ it is then. Pretty monstrous (to a non-set-theory aficionado at least). That's a very nice last equation you wrote. Does it have a special name? Where in standard books would I find such wonderful gems? – joshphysics Oct 19 '14 at 18:16
  • I don't know where you can find the last equation. It's not hard to prove if you know basic cardinal arithmetic. I don't know if it has a name either. I wrote a proof in an answer before, I'll go find it. – Asaf Karagila Oct 19 '14 at 18:21
  • There we go: http://math.stackexchange.com/a/194287/622 – Asaf Karagila Oct 19 '14 at 18:28
  • Wonderful! Thanks Asaf. – joshphysics Oct 19 '14 at 18:30