Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $$C^1_0[a,b]:=\{f \ C^1[a,b]|f(a)=f(b)=0\}.$$ Providing $C^1_0[a,b]$ is dense in $L^2[a,b]$, I want to prove the following statement:

if for $g,h\in L^2[a,b]$,

$$\int_a^b g \phi \,dx =\int_a^b h \phi \,dx$$ for all test functions $\phi\in C^1_0[a,b]$,

then $g = h$ almost everywhere.

It seems to be similar with fundamental lemma of calculus of variation, how can I extend the the result to $L^2$ functions? My guess is using density argument, along with one of convergence theorems, but I failed to construct the proof.

Please help me out here. Thank you.

share|cite|improve this question
up vote 2 down vote accepted

You can do this abstractly. Show that the orthogonal complement of $g-h$ is the whole $L^2$ space. By Hilbert space theory this implies that $g-h$ is the null vector, that is, $g=h$ almost everywhere.

share|cite|improve this answer
Since $g-h$ is orthogonal to all $\phi \in C^1_0[a,b]$, so $g-h$ is orthogonal to $\bar{C^1_0[a,b]}=L^2[a,b]$? Is that correct? – newbie Dec 18 '12 at 0:38
@newbie: Exactly. – Giuseppe Negro Dec 18 '12 at 0:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.