1.1 Normed Linear Spaces

Question 1

What is a norm?

Answer 1

A norm on a linear space is a real value function ||.|| with the following axioms:
1. ||x|| >= 0
2. ||x|| = 0 iff x = 0
3. ||alpha x|| = |alpha| ||x||
4. ||x + y|| <= ||x|| + ||y||

Question 2

What is normed linear space?

Answer 2

It is the pair of the norm, together with the linear space (X, ||.||), following the axioms of a norm.

Question 3

What are some examples of normed linear spaces?

Answer 3

1. The linear space R has the norm:
   || . || : R -> R
   ||x|| = |x|