If we define derivation on $C(X)$, where $X$ is a compact Hausdorff space, will the dimension of $Der(C(X))$(if it is well-defined) equal to the (topological) dimension of $X$? My guess is this is true since derivation is just differentiation of function, but I cannot prove it.
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1$\begingroup$ $C(X)$ consists of continuous functions so there are not obviously any nonzero derivations on $C(X)$ at all. $\endgroup$Qiaochu Yuan– Qiaochu Yuan2021-01-22 20:52:07 +00:00Commented Jan 22, 2021 at 20:52
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$\begingroup$ @QiaochuYuan its not entirely clear to me what your comment means (I can't parse it) $\endgroup$s.harp– s.harp2021-01-22 21:18:35 +00:00Commented Jan 22, 2021 at 21:18
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1$\begingroup$ I mean you say "derivation is just differentiation of function" and that argument would apply to the smooth functions $C^{\infty}(M)$ on a smooth manifold, but looking at just the continuous functions on a compact Hausdorff space there is no obvious notion of differentiation available (even if that compact Hausdorff space happens to be a manifold). $\endgroup$Qiaochu Yuan– Qiaochu Yuan2021-01-22 21:26:35 +00:00Commented Jan 22, 2021 at 21:26
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Suppose $D$ is a derivation and $f\in C(X)$. For $x\in X$ look at $(Df)\,(x)$, since $D$ applied to any constant function is necessarily $0$ you have $$(Df)\,(x) = D(f-c )\, (x)$$ with $c$ a constant function. Now for $c=f(x)$ you have that $f-c$ is $0$ at $x$. You may check that there exist $g,h$ continuous with $f-c= g\cdot h$ and $g(x)=h(x)=0$ (this is a feature of continuous functions that fundamentally fails for any notion of differentiable functions). Then
$$(Df)\, (x) = D(f-c)\,(x) =D(g\cdot h)\,(x)= g(x) (Dh)\,(x)+(Dg)\,(x)\,h(x) =0$$
hence $Df$ is zero at $x$ for any $x\in X$. This implies $D=0$ so there are no non-zero derivations.
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$\begingroup$ May you further explain the property that $f-c=gh$, and why it fails on differentiable function? $\endgroup$Ken.Wong– Ken.Wong2021-01-23 11:42:14 +00:00Commented Jan 23, 2021 at 11:42
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1$\begingroup$ Consider the function $x\mapsto x$ on $[-1,1]$, which is zero at $0$. Can you find two continuous functions $g,h$ with $x= g(x)h(x)$ and $g(0)=0=h(0)$? What about for differentiable $g,h$? Once you have understood how to get $g,h$ for this example you can find the expression for an aribtrary $f$ rather easily. $\endgroup$s.harp– s.harp2021-01-23 13:16:28 +00:00Commented Jan 23, 2021 at 13:16