Assume that f is a continuous function on $[a,b]$ such that for any continuous function g on $[a,b]$ $\int_a^b f(x)g(x)dx = 0$, then how can I show that f(x) = 0 for all $x\in [a,b]$?
Asked
Active
Viewed 3,885 times
3
-
See also: http://math.stackexchange.com/questions/1128621/how-can-i-show-that-f-must-be-zero-if-int-fg-is-always-zero – Martin Sleziak Feb 19 '15 at 10:52
-
This question is at the moment closed as a duplicate of Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$. While answers to that question answer this one too, I think that this question can be solved in a much easier way. Maybe How can I show that $f$ must be zero if $\int fg$ is always zero? would be a better duplicate target. – Martin Sleziak Feb 20 '15 at 06:10
3 Answers
7
Hint: What happens if $g(x)=f(x)$? Since $g$ can be any continuous function, beginning with this is okay.
Clayton
- 24,751
4
What happens if you take $g=\bar{f}$? In case $f$ is complex-valued. If it is real-valued, this becomes $g=f$, of course.
Julien
- 44,791
4
Write $g = f$. Then $g(x)\cdot f(x)=f(x)^2\geq 0$ for all $x\in[a,b]$. Deduce the result of $$ \int_a^bf(x)^2 \, dx=0. $$ If exist $x_o\in[a,b]$ such that $f(x_o)>0$ and $f$ continuous then $$ \int_a^bf(x)^2 \, dx>0. $$
Elias Costa
- 14,658