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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 continuous functio...
https://math.stackexchange.com/questions/1344706/are-continuous-functions-with-compact-support-bounded?noredirect=1
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 continuous …
https://www.ams.org/journals/tran/1971-156-00/S0002-9947-1971-0275367-4/S0002-9947-1971-0275367-4.pdf
1. Introduction. The support of a real continuous function / on a topological space A" is the closure of the set of points in Afat which/does not vanish. Gillman and Jerison have shown that when A'is a realcompact space, the functions in C(X) with compact support are precisely the functions which belong to every free maximal ideal in C(X).
https://en.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space
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.
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 …
https://en.wikipedia.org/wiki/Function_space
() continuous functions with compact support bounded functions continuous functions which vanish at infinity continuous functions that have continuous first r derivatives.
https://en.wikipedia.org/wiki/Bounded_function
This function can be made bounded if one considers its domain to be, for example, [2, ∞) or (−∞, −2]. The function defined for all real x is bounded. Every continuous function f : [0, 1] → R is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
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://en.wikipedia.org/wiki/Compact_space
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.) As a sort of converse to the above statements, the pre-image of a compact space under a proper map is compact.
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