Is this a metric on $C[0,1]$?

Question

At $C[0,1]$ we define the function $$d(f,g)=\int_{0}^{1}\vert(f(t)-g(t))(2f(t)+3g(t))\vert dt. $$ Is $d$ a metric on $C[0,1]$?

Isn't f(t) =2, g(t) =1 an obvious counterexample to the symmetry? — Boxonix, Jan 11 '20 at 17:02

score 3 · Accepted Answer · answered Jan 11 '20 at 17:01

3

It is not a metric, since there are two different functions $f,g$ such that $d(f,g)=0$

For example $$f(x)=3x,g(x)=-2x$$

answered Jan 11 '20 at 17:01

TheHolyJoker

1

thanks I see where exactly I was wrong in the first part – Kostas Jan 11 '20 at 17:06
@Kostas Please consider accepting one of the answers provided. How to accept an answer – TheHolyJoker Jan 15 '20 at 17:42

score 2 · Answer 2 · answered Jan 11 '20 at 17:02

2

Take $f \equiv 2, g \equiv -1$

Then $d(f,g) = \int_{0}^{1} 3 dt = 3$

But $d(g,f) = \int_{0}^{1}12dt = 12$

So is not symmetric

answered Jan 11 '20 at 17:02

ZAF

2 Answers2