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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 , …
https://math.stackexchange.com/questions/1344706/are-continuous-functions-with-compact-support-bounded
While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded …
http://www.msc.uky.edu/ken/ma570/lectures/lecture2/html/compact.htm
Theorem 5: (Heine-Borel Theorem) With the usual topology on , a subset of is compact if and only if it both closed and bounded. Note: The Extreme Value Theorem follows: If is continuous, then is the image of a compact set and so is compact by Proposition 2. So, it is both closed and bounded by Exercise 5.
https://en.wikipedia.org/wiki/Bounded_function
Every continuous function f : [0, 1] → R is bounded. More generally, any continuous function from a compact space into a metric space is bounded. All complex-valued functions f : C → C which are entire are either unbounded or constant as a consequence of Liouville's theorem. In particular, the complex sin : C → C must be unbounded since ...
https://en.wikipedia.org/wiki/Compact_space
Properties of compact spaces Functions and compact spaces. A continuous image of a compact space is compact. This implies the extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum. (Slightly more generally, this is true for an upper semicontinuous function.)
https://en.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space
In the non-compact case, however, C(X) is not in general a Banach space with respect to the uniform norm since it may contain unbounded functions. Hence it is more typical to consider the space, denoted here C B ( X ) of bounded continuous functions on X .
https://en.wikipedia.org/wiki/Bump_function
Examples. The function : → given by = { (− −), ∈ (−,),is an example of a bump function in one dimension. It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded and closed support.
https://mathoverflow.net/questions/159853/rieszs-representation-theorem-for-non-locally-compact-spaces
Riesz's representation theorem for non-locally compact spaces ... recent statement concerning the dual of the algebra of bounded continuous functions on non-locally-compact spaces? What is lost when one gives up local-compactness? (Please notice that I am not interested in the algebra of functions with compact support or vanishing at infinity.) ...
https://www.math.ucdavis.edu/~hunter/m127c/hmwk6_solutions.pdf
If f, g are continuous functions with compact support, prove that kf ∗gk 1 ≤ kfk 1kgk 1. Solution. • (a) If f, g are continuous functions with compact support, then they are bounded and uniformly continuous, since the functions are zero outside a compact (i.e. closed, bounded) interval, and a continuous
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