xamples -X = R with the norm ||x|| = |x| x € R is normed space

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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2.4. Normed Space
A norm on a (real or complex) vector space X is a real-valued function on
at an x E X is denoted by ||x|| (read "norm of x") and which has the pro
(NI) ||x|| > 0
(N2) ||x|| = 0 + x = 0,
(N3) || Ax|| = |2|||x||
(N4) ||x + y|| < ||x|| + |ly|| Vx,y € X (Triangle inequality).
V x E X,
(Vx E X ,V 1 e K),
Definition. A normed space is a vector space X with a norm defined on
Remark . The normed space just defined is denoted by (X, ||. || ) or simply
Examples
1) X = R with the norm ||x|| = |x| x E R is normed space
Transcribed Image Text:2.4. Normed Space A norm on a (real or complex) vector space X is a real-valued function on at an x E X is denoted by ||x|| (read "norm of x") and which has the pro (NI) ||x|| > 0 (N2) ||x|| = 0 + x = 0, (N3) || Ax|| = |2|||x|| (N4) ||x + y|| < ||x|| + |ly|| Vx,y € X (Triangle inequality). V x E X, (Vx E X ,V 1 e K), Definition. A normed space is a vector space X with a norm defined on Remark . The normed space just defined is denoted by (X, ||. || ) or simply Examples 1) X = R with the norm ||x|| = |x| x E R is normed space
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