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I see $C[0,1]$, the set of continuous functions $f : [0,1] \to \mathbb{R}$ used a lot as an example of an infinite-dimensional vector space. There are other examples, too. But I notice that I've never seen the set $S$ of all functions $f : [0,1] \to \mathbb{R}$ or the like used; there's always some kind of continuity/differentiability requirement given along with it. So is $S$ a vector space?

  • $f + g$ and $cf$ are in $S$ for $f,g \in S$
  • There is an additive identity $f=0$, and additive inverse $-f$, and function addition is commutative and associative
  • The rest of the axioms are just as easy to verify: $1f=f$, $a(bf)=(ab)f$, $(a+b)f=af+bf$, and $a(f+g)=af+ag$

So I conclude that $S$ is a vector space. But why do we not see it mentioned? Is it because you can't in general integrate an element of $S$, or because there are other important structures that $S$ is not but $C[0,1]$ is? Have I made a mistake somewhere and $S$ is not actually a vector space? Are we biased towards continuity and $S$ isn't as useful or interesting? Am I just not looking hard enough?

Math2718
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    It's a fine vector space. It's a little too "big" to do much with. – Randall Jun 10 '20 at 13:59
  • In fact, given any set $X$, the set of functions $f:X\to \Bbb R$ is a fine vector space. (In fact, we can replace $\Bbb R$ by any vector space $V$ with appropriate definition of sum of functions and scalar multiplication.) – Aryaman Maithani Jun 10 '20 at 14:00
  • Also, there's nothing special about $[0,1]$, or even subsets of the reals. You can make the set of all functions $f: X \to \mathbb{R}$ into a vector space in the pointwise way, where $X$ can be any set at all. – Randall Jun 10 '20 at 14:00
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    $C$ is a Banach space with the sup-norm (and it can be proven easily, in the sense that it's usually done in real analysis, before functional analysis), while $S$ is not. You can easily equip $C$ with a scalar product (because it's a subspace of $L^2$) and make it a pre-Hilbert space, while you can't do it with $S$. – Botond Jun 10 '20 at 14:06
  • Just checked today's morning, but this vector space is indeed shown in the Algebra courses in my university. Coincidence. – dEmigOd Jun 10 '20 at 14:11
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    It's certainly a vector space; it's just not as interesting as something like $C([0, 1])$. In the latter, for example, there's a mild amount of work to be done to show it's a vector space (which textbooks care about); there's something interesting but tractable to say about the dual; it arises naturally in later contexts; and it has more interesting things to talk about with respect to norms, completeness, and so on. They're both perfectly reasonable, examples, though (and I have seen both used as examples, so maybe you'd prefer a different textbook than whichever one you're using). – anomaly Jun 10 '20 at 15:10
  • Does this answer your question? Vector space of a Function (Example for understanding the concept) – Michael Hoppe Jun 10 '20 at 17:53
  • @MichaelHoppe my question is not about what vector spaces are, it's about why I had never seen $S$ as an example of a vector space. The comments given by Randal, Botond, and anomaly answered the question. – Math2718 Jun 10 '20 at 20:15
  • Why did you unaccept my answer? You're entitled of course, but I don't see the reason. – Matt Samuel Jun 17 '20 at 20:23
  • @MattSamuel I feel like the answer I was looking for was just that $S$ is too large to do stuff with or other reasons why $S$ might not be interesting. Or to be told that $S$ is interesting, if it is. I feel like the comments on this post answered the question, and so I accepted the answer as a way to say my question was answered. But tbh, the answer wasn't quite answer that I was looking for. I feel pretty confused as to what I should do in this scenario. – Math2718 Jun 18 '20 at 00:50

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