I am having trouble with a homework problem. This is the problem:

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify at least one of the ten vector space axioms that fails. $\mathcal C\,[-1,1]$, the set of all continuous functions defined on the interval $[-1,1]$, with the standard operations.

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    $\begingroup$ Do you know what the axioms for a vector space is? $\endgroup$ – Brian Fitzpatrick Feb 14 '14 at 5:44
  • $\begingroup$ What exactly is the problem? $\endgroup$ – Asaf Karagila Feb 14 '14 at 5:45
  • $\begingroup$ Yes, I have the list of all ten of them. $\endgroup$ – Elizabeth Vemmer Feb 14 '14 at 5:47
  • $\begingroup$ The problem, Asif, is that I'm how to work this. I must prove that it either is, or isn't a vector space. Essentially, if I cannot "break" an axiom with it. $\endgroup$ – Elizabeth Vemmer Feb 14 '14 at 5:48
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    $\begingroup$ You should be clear what $+$ and $\cdot$ mean on the space $C[-1,1]$. $\endgroup$ – copper.hat Feb 14 '14 at 6:33

Any continuous function from a connected space to a totally disconnected space must be constant, and thus any continuous function from $[-1,1]$ to a totally disconnected vector space $V$ (such as a finite-dimensional vector space over $\mathbb F_p$ or a number field, with the topology induced by the base field) is constant. So the set of continuous functions from $[-1,1]$ to $V$ is in bijection with $V$ via the map $f\mapsto f(0)$ and thus can be endowed with its vector space structure (induced by the bijection), so $\mathcal{C}[-1,1]$ is a vector space.

I'll leave the more difficult case of connected fields for another user.


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