Answer to Question #126922 in Functional Analysis for sana khyzer

Question #126922

If a sub additive functional defined on a normed space X is nonnegative outside a sphere {x Illxll = r}, show that it is nonnegative for all x E X


1
Expert's answer
2020-07-21T18:26:24-0400

Suppose that "f" is a sub additive functional on "X". Then, by definition, we have

"\\forall x,y \\in X \\quad f(x+y)\\leq f(x)+f(y)." For any "x\\in X" outside a sphere "f(x)\\geq0" . Suppose that "\\tilde{x}" lies either inside a sphere or on its boundary. This means that "||\\tilde{x}||=\\alpha\\leq r". We consider two cases:

  1. "\\alpha>0" . We set "n=[\\frac{r}{\\alpha}]+2" , where "[]" denotes the truncation. Then "n\\alpha>r" . It means, that "||n\\tilde{x}||>r" and "n\\tilde{x}" lies outside the sphere. Using sub additivity, we get "0\\leq f(n\\tilde{x})=f((n-1)\\tilde{x}+\\tilde{x})\\leq f(\\tilde{x})+f((n-2)\\tilde{x}+\\tilde{x})\\leq...\\leq n\\,f(\\tilde{x})"

Therefore, we have "f(\\tilde{x})\\geq0" .

2. "\\alpha=0" . Using the definition of the norm, we have "\\tilde{x}=0" . Assume that "x\\in X" is outside the sphere. Then, "f(x+0)\\leq f(0)+f(x)." From the latter we get "f(0)\\geq0" .

Thus, "f(x)\\geq0" for all "x\\in X."


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Comments

Assignment Expert
22.07.20, 23:02

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sana khyzer
22.07.20, 10:22

Awsomeeeee

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