# Let $F$ the set of all continuous real functions with domain $[0,a]$. Which of the following are metrics on $F$?

Let $$F$$ the set of all continuous real functions with domain $$[0,a]$$. Which of the following are metrics on $$F$$?

1. $$d(f_1,f_2)$$ is the maximum value of $$|f_1(x)-f_2(x)|$$ for $$x\in[0,a],$$
2. $$d(f_1,f_2)=\int^a_0|f_1(x)|-|f_2(x)|$$
3. $$d(f_1,f_2)=\int^a_0 |f_1(x)-f_2(x)|$$
4. $$d(f_1,f_2)=\int ^a_0|f_1(x).f_2(x)|$$

My attempt:

for (3)

(1) $$d(f_1,f_2)\ge 0\; \forall x\in [0,a]$$

(2) $$d(f_1,f_2)=0 \iff \int ^a_0|f_1(x)-f_2(x)| \iff|f_1(x)-f_2(x)|=0 \iff f_1(x)=f_2(x)$$

(3) $$d(f_1,f_2)=\int^a_0 |f_1(x)-f_2(x)|=\int^a_0 |f_2(x)-f_1(x)|=d(f_2,f_1)$$

(4) $$d(f_1,f_2)=\int^a_0 |f_1(x)-f_2(x)|\le d(f_1,f_2)=\int^a_0 |f_1(x)-f_3(x)|+ \int^a_0 |f_3(x)-f_2(x)|\le d(f_1,f_3)+d(f_3,f_2)$$

So (3) is metric

(4) is not metric since it not satisfies property $$d(f_1,f_2)=0\iff f_1=f_2$$

However, I am not sure about that.

Can any one help with (1) and (2)?

(1) is a metric. Just check that it has all the properties of a metric.

(2) isn't. For instance, if $f_1$=0 and $f_2=1$, then $d(f_1,f_2)<0$.

• ....thank your fast response........is my reason for (4) is true sir?.....@José Carlos Santos – Inverse Problem Apr 11 '18 at 10:15
• @SureshPonnada Yes, you are right about (4). But you should have added an example ($f_1=0$ and $f_1=1$, for instance). – José Carlos Santos Apr 11 '18 at 10:16
• thank you sir once again .......@José Carlos Santos – Inverse Problem Apr 11 '18 at 10:17