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The other one, the topology introduced by schwartz, is complete (in a generalized sense) but not metrizable, and is the one used to define distributions. In neither case C_0^\infty is a
Frechet space. Of course one can also think to that space with the sup norm but that is quite unnatural since immediately afterwards the article claims its dual is the space of distributions, so i argue we are considering the schwartz topology.
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734:{\displaystyle g(x)={\frac {1+\tanh \sinh(2\tanh ^{-1}(|x|-|x-1|))}{2}}={\begin{cases}0&x\leq 0\\{\frac {1+\tanh \sinh(2\tanh ^{-1}(2x-1))}{2}}&0\leq x\leq 1\\1&1\leq x\end{cases}}={\begin{cases}0&x\leq 0\\{\frac {1}{1+e^{\frac {2x-1}{x^{2}-x}}}}&0<x<1\\1&1\leq x\end{cases}}}
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The statement "C^infty_0 is a frechet space" is meaningless unless we specify the topology we are using; indeed interesting topologies on this space does not make it into a frechet space. Usually two topologies are used: the first one, the uniform convergence topology, is metrizable but not complete.
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I'm not very expert C^\infty_0 may or may not be
Frechet space, depending upon the topology we are considering...for example with the uniform convergence topology it is not metrizable, and that is the topology used to define distributions, so is misleading to say that c^\infty_o is a frechèt space.
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Correct me if I'm wrong, but the piecewise example in the article is not continuous, and does not match the graph. By l'hopital, the limit at 1 or -1 is one, because "1/(x^2-1)" converges to 0 at 1. (The limit equals "0/2x = 0"). This can be fixed by simply subtracting 1 from that secton of the
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of n-dimensional bump functions is closed not only under addition and scalar multiplication, but also under the multiplication of its elements by each other. This makes it into an algebra. Shouldn't this be mentioned in the
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Are these things that could/should be added to the article, or would that be problematic/unhelpful?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Rendering of Bump function expression in Chrome on
Windows appears wrong
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Nevermind. 1/(x^2-1) is not indeterminate, so l'hopital doesn't apply.
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Isn't the space of n-dimensional bump functions actually an algebra?
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272:{\displaystyle C_{0}^{\infty }(\mathbf {R} ^{n})}
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