A seminorm on a vector space X is a real-valued function p on X such that p(x + y) s p(x) +p(y) p(ax) = |a|p(x) for all x, y E X and all scalar a. i. ii. Show that the following properties for seminorm: a) p(0) = 0. b) Ip(x) – p(y)| s p(x - y)

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this question  is from course functional analysis kindly solve it with explanation.

A seminorm on a vector space X is a real-valued function p on X such that
p(x + y) < p(x) +p(y)
ii.
p(ax) = |a|p(x) for all x, y E X and all scalar a.
i.
Show that the following properties for seminorm:
a) p(0) = 0.
b) Ip(x) – p(y)| s p(x - y)
Also show that the mapping x If(x)| is seminorm. Discuss and prove that the conditions when
seminorm becomes norm.
Transcribed Image Text:A seminorm on a vector space X is a real-valued function p on X such that p(x + y) < p(x) +p(y) ii. p(ax) = |a|p(x) for all x, y E X and all scalar a. i. Show that the following properties for seminorm: a) p(0) = 0. b) Ip(x) – p(y)| s p(x - y) Also show that the mapping x If(x)| is seminorm. Discuss and prove that the conditions when seminorm becomes norm.
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