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bounded function
Definition Suppose is a nonempty set. Then a function is a bounded function if there exist a such that for all . The set of all bounded functions on is usually denoted by ([1], pp. 61).
Under standard point-wise addition and point-wise multiplication by a scalar, is a complex vector space.
If , then the sup-norm, or uniform norm, of is defined as
It is straightforward to check that makes into a normed vector space, i.e., to check that satisfies the assumptions for a norm.
0.0.1 Example
Suppose is a compact topological space. Further, let be the set of continuous complex-valued functions on (with the same vector space structure as ). Then is a vector subspace of .
References
- 1 C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, 2nd ed., Academic Press, 1990.
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new image: approximation_to_sinx by jeremyboden
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new image: LineDiagrams.jpg by m759
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May 16
new problem: Curve fitting using the Exchange Algorithm. by jeremyboden
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