Q. Prove that normed linear space is also a metric space.

Question

This question is about Functional analysis.

Answer 1

Linear space is a number line. To qualify as a metric space the following must be true Ɐ x,y,z ∊ ℝ.

1. The absolute distance between two points can be represented by d(x,y)=|x-y|.
2. d(x,y)=d(y,x)=|x-y|. Example x=3 y=10; d(3,10): x-y=-7; d(10,3): y-x=7; |-7|=|7|=7.
3. d(x,y)=0 iff x=y. Example x=y=3, d(x,y)=3-3=0.
4. triangular inequality: d(x,z)+d(z,y)≥d(x,y), example x=1, y=4, z=7: 6+3≥3 true; example x=1, y=7, z=4: 3+3≥6 true.

Therefore linear space is a metric space.