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http://mathworld.wolfram.com/CompactSupport.html
Jan 02, 2020 · A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For example, the function f:x->x^2 in its entire domain (i.e., f:R->R^+) does not have compact support, while any bump function does have compact support.
https://math.stackexchange.com/questions/1425758/space-of-smooth-functions-with-compact-support
The support of a function is closed, by definition. Every closed subset of a compact set is compact. So, if ω is a compact set, the subscript is pointless. Just write C∞(ω) then. Better yet, write C∞(K) because the letter K is typically associated with compact sets, while ω is associated with open sets.
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.
https://www.planetmath.org/CompactlySupportedContinuousFunctionsAreDenseInLp
We denote by C c (X) the space of continuous functions X → ℂ with compact support. Theroem - For every 1 ≤ p < ∞, C c (X) is dense in L p (X) (http://planetmath.org/LpSpace).
https://en.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by C(X), is a vector space with respect to
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.
https://en.wikipedia.org/wiki/Distribution_(mathematics)
Test function space. The space D(U) of test functions on U is defined as follows. A function : U → R is said to have compact support if there exists a compact subset K of U such that (x) = 0 for all x in U \ K.
https://math.rice.edu/~semmes/fun5.pdf
Now let X be a compact topological space, and let C(X) be the space of real or complex-valued continuous functions on X, equipped with the supremum norm. In particular, C(X) is a metric space with respect to the supremum metric, and it is well known that C(X) is complete.
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