quasar987

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**[SOLVED] A caracterisation of f=0 by integrals**

**1. Homework Statement**

Does anyone know how to show, or know a book that proves the implication

[tex]\left(\int_0^1f\varphi = 0 \ \ \forall \varphi \in C_c^1([0,1])\right)\Rightarrow f=0[/tex]

for f in L²([0,1]) and where [itex]C_c^1([0,1])[/itex] denotes the C^1([0,1]) functions whose support is contained in (0,1).

Thanks.

**3. The Attempt at a Solution**

I tried using the density of [itex]C_c^1([0,1])[/itex] in L^1 to obtained a sequence [itex]\varphi_n[/itex] that converges pointwise a.e. to the caracteristic function of [0,1] and then plugging-in the convergence theorems (Fatou, motone and dominated) but I eventually aknowledged that this would not work. Well, at least I got

[tex]\int_0^1f\leq 0[/tex]

out of Fatou.