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https://math.stackexchange.com/questions/220590/the-subset-of-c-infty-functions-with-compact-support-in-mathbbr-in-the
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https://www.quora.com/Are-Bump-functions-C-infty-functions-with-compact-support-nowhere-analytic
By [math]C-\infty[/math], I assume you mean a function [math]f:\mathbb C\to\mathbb C[/math] with derivatives of all orders. The idea of compact support is clear — all the action is taking place on some compact subset [math]K[/math] of [math]\mathbb C[/math].
https://en.wikipedia.org/wiki/Smoothness
A bump function is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous. A smooth function is a function that has derivatives of …
https://web.math.princeton.edu/~const/spa.pdf
of continuous functions on a compact is C(K) = ff: K!Cjfcontinuousg where KˆRn is compact. The norm is kfk= sup x2K jf(x)j. The H older class C is the space of bounded contuous functions with norm kfk C = sup x2 jf(x)j+ sup x6=y jf(x) f(y)j jx yj with 0 < <1. When = 1 we have the Lipschitz class. We will describe Sobolev classes shortly.
http://www-users.math.umn.edu/~garrett/m/fun/notes_2016-17/examples.pdf
Paul Garrett: Examples of function spaces (February 11, 2017) converges in sup-norm, the partial sums have compact support, but the whole does not have compact support. [2.1] Claim: The completion of the space Co c (R) of compactly-supported continuous functions in the metric given by the sup-norm jfj Co = sup x2R jf(x)jis the space C o
https://en.wikipedia.org/wiki/Support_(mathematics)
In good cases, functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example.
https://en.wikipedia.org/wiki/Function_space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication.
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