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https://math.stackexchange.com/questions/242877/compact-support-functions-dense-in-l-1
Another approach uses the representation of the dual [L1(R)] ⋆ as L∞(R) and the Hahn-Banach separation theorem. Namely, to prove that a vector subspace of a Banach space is dense we only need to show that the only continuous linear functional that vanishes on it is the null one. To do this fix ϕ ∈ L∞(R) and suppose...
https://www.planetmath.org/CompactlySupportedContinuousFunctionsAreDenseInLp
Now, it follows easily that any simple function ∑i=1nciχAi, where each Ai has finite measure, can also be approximated by a compactly supported continuous function. Since this kind of simple functions are dense in Lp(X) we see that Cc(X) is also dense in Lp(X).
https://mathproblems123.files.wordpress.com/2011/02/density-1.pdf
Oct 03, 2004 · 1 Approximation by continuous functions In this supplement, we’ll show that continuous functions with compact support are dense in L1 = L1(Rn;m). The support of a complex valued function f on a metric space X is the closure of fx 2 X : f(x) 6= 0g. We’ll denote by Cc(X) the set of all complex valued continuous functions on X with compact support.
https://mathoverflow.net/questions/267710/continuous-functions-dense-in-l-1
If X is a complete doubling metric space equipped with a complete probability measure μ such that all Borel sets are μ -measurable, then Cc(X) --- the continuous functions with compact support --- are dense in L1(μ). Question: What are the weakest conditions under which Cc(X) is dense in L1(μ)...
http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/Chapter%2011-%20Convolutions%20and%20Approximations.pdf
Proposition 11.6. Let (X,τ) be a second countable locally compact Hausdor ff space, BX = σ(τ) be the Borel σ—algebraandµ: BX →[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,∞). Proof. First Proof.
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-algebra of M.Let S(A) denote those simple functions φ: X→C such that
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.nthu.edu.tw/~kchen/teaching/5131week3.pdf
dense in Lp(E). These step functions are linear combinations of characteristic functions on some dyadic cubes. This implies that the space of simple functions is also dense in Lp(Rn). In this section we prove that the space of smooth functions with compact supports, and the space of functions with rapidly decreasing derivatives are also dense ...
https://en.wikipedia.org/wiki/Locally_integrable_function
In mathematics, a locally integrable function is a function which is integrable on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain: in other words, locally integrable functions can grow arbitrarily fast at the …
http://home.iitk.ac.in/~chavan/afs_almora.pdf
The vector space of continuous functions with compact support is dense in L1: Proof. Let f 2L1 and >0 be given. By Problem 3.1, there exists a simple function ssuch that kf sk< =3:Since sis a nite linear combination of char-acteristic functions, by the preceding problem, there exists a step function gsuch that ks gk 1 < =3:Now by Problem 3.3 ...
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