Normal linear space


Definition- let X be a linear space over the field of scalar k and let || .|| be a mapping from X into R such that for all x ,y £X and all α £ C(or) R, we have
(1) ||x|| >0 if x ≠ 0
||x||=0 iff x=0
(2) ||α x|| =|α| ||x||
(3) ||x +y||≤ ||x|| +||y||
Then the function || . || is called a norm on X and the pair (X, || . || is called a complex (or real)
normal linear space.
EXAMPLE – The real line R over R is a normal linear space with the norm || .|| defined by || x|| =|x| for all x £ R The complex plane C is a real linear space (C,R) and the mapping Z →|Z| is a norm on (C,R)

Author:- Megha Sharma
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