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https://www.planetmath.org/CompactlySupportedContinuousFunctionsAreDenseInLp
compactly supported continuous functions are dense in L p. ... We denote by C c (X) the space of continuous functions X → ℂ with compact support. Theroem - For every 1 ... By of μ, we know there exist an open set U and a compact set K such that K ...
https://math.stackexchange.com/questions/8504/are-the-smooth-functions-dense-in-either-mathcal-l-2-or-mathcal-l-1
Yes. In fact, by the Stone-Weierstrass theorem and the existence of smooth bump functions, smooth functions with compact support are uniformly dense in the space of continuous functions with compact support.
http://www.math.nthu.edu.tw/~kchen/teaching/5131week3.pdf
4. DENSE SUBSPACES OF Lp 127 4. Dense Subspaces of Lp In the proof of Theorem 3.4 we constructed a countable collection of step functions which is dense in Lp(E).These step functions are linear combinations of characteristic functions on some
https://mathoverflow.net/questions/237636/are-compactly-supported-continuous-functions-dense-in-the-continuous-functions-o
Continuous functions on $\mathbb R^d$ such that the support is a compact subset of $\overline{\Omega}$? For "nice" $\Omega$ this would be the space of continuous functions on $\Omega$ vanishing at the boundary. $\endgroup$ – Jochen Wengenroth Apr 29 '16 at 12:50
http://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/chap22.pdf
22 Approximation Theorems and Convolutions 22.1 Density Theorems In this section, (X,M,µ) will be a measure space A will be a subalgebra of M. Notation 22.1 Suppose (X,M,µ) is a measure space and A ⊂M is a sub- ... of continuous functions with compact support) is dense in Lp ...
http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/Chapter%2011-%20Convolutions%20and%20Approximations.pdf
Approximation Theorems and Convolutions ... Let (X,τ) be a second countable locally compact Hausdor ff ... 0,∞] be a measure such that µ(K) <∞when Kis a compact subset of X.Then Cc(X) (the space of continuous functions with compact support) is dense in Lp(µ) for all p∈[1,∞).
http://math.arizona.edu/~faris/realb.pdf
Consider now the special case when Xis a locally compact Hausdor space. Thus each point has a compact neighborhood. For example X could be Rn. The space Cc(X) consists of all continuous functions, each one of which has compact support. The space C0(X) is the closure of Cc(X) in BC(X). It is itself a Banach space.
https://www.encyclopediaofmath.org/index.php/Function_of_compact_support
The support of is the closure of the set of points for which is different from zero . Thus one can also say that a function of compact support in is a function defined on such that its support is a closed bounded set located at a distance from the boundary of by a number greater than , where is sufficiently small.
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