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https://math.stackexchange.com/questions/67370/smooth-functions-with-compact-support-are-dense-in-l1
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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. In other scenarios, the function space might ...
https://en.wikipedia.org/wiki/Bump_function
In mathematics, a bump function (also called a test function) is a function: → on a Euclidean space which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported.The set of all bump functions with domain forms a vector space, denoted ∞ or ∞ ().The dual space of this space endowed with a suitable topology is the space of distributions
https://www.planetmath.org/SmoothFunctionsWithCompactSupport
Then the set of smooth functions with compact support (in U) is the set of functions f: ℝ n → ℂ which are smooth (i.e., ∂ α f: ℝ n → ℂ is a continuous function for all multi-indices α) and supp f is compact and contained in U. This function space is denoted by C 0 ∞ (U).
https://en.wikipedia.org/wiki/Smooth_function
In this way smooth functions between manifolds can transport local data, like vector fields and differential forms, from one manifold to another, or down to Euclidean space where computations like integration are well understood. Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions.
https://en.wikipedia.org/wiki/Compact_support
Functions with compact support on a topological space are those whose closed support is a compact subset of . If X {\displaystyle X} is the real line, or n {\displaystyle n} -dimensional Euclidean space, then a function has compact support if and only if it has bounded support , since a subset of R n {\displaystyle \mathbb {R} ^{n}} is compact ...
https://en.wikipedia.org/wiki/Distribution_(mathematics)
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if A : D(U) → D(U) is a linear mapping of vector spaces which is continuous with respect to the weak-* topology, then it is possible to extend A to a mapping A : …
https://math.rice.edu/~semmes/fun5.pdf
Some basic aspects of smooth functions and distributions on open subsets of Rn are briefly discussed. Contents 1 Smooth functions 2 2 Supremum seminorms 3 3 Countably many seminorms 4 4 Cauchy sequences 5 5 Compact support 6 6 Inductive limits 8 7 Distributions 9 8 Differentiation of distributions 10 9 Multiplication by smooth functions 11
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://mathoverflow.net/questions/166029/is-the-space-of-smooth-functions-with-compact-support-a-df-space
It is not clear to me what the second question of your title means but as regards the one in the text, the space of smooth functions with compact support has, of course, as a complete lcs, a canonical representation as a projective limit of Banach spaces but the seminorms involved are rather complicated.
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