Metric Space

Jan 2009
145
37
I'm trying to show that, given X is the set of all continuous functions \(\displaystyle f:[a,b] \rightarrow R \), and d(f,g) = \(\displaystyle \int_{a}^{b} | f(t) - g(t) | dt \), that (X, d) defines a metric space.

My confusion is coming in when I let a = 0, b = \(\displaystyle \frac{\pi}{2} \), and f = sine and g = cosine. Then wouldn't the distance function between these two be

\(\displaystyle \int_{0}^{ \frac{\pi}{2}} | sin(t) - cos(t) | dt \)
\(\displaystyle = - (cos(\frac{\pi}{2}) - cos(0)) - ( sin(\frac{\pi}{2}) - sin(0)) \)
\(\displaystyle = -(0 - 1) - (1 - 0) = 1 - 1 = 0 \)

But f != g, so isn't that a problem?
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
What did you make of the absolute value?

Your calculation is wrong. The integral of a non-negative continuous function is 0 if and only if the function is identically 0. The function you are integrating is clearly non-negative and continuous, but it's not identically 0!