# real analysis – Prove that \$int_0^1g=sum_{j=1}^Nlambda_jg(x_j)\$,\$forall gin V\$.

Suppose $$Vsubset C(0,1)$$is a linear space and $$dim V=N$$,and there exists an element $$uin V$$,such that $$int_0^1uneq 0$$.Suppose $$Omega subset (0,1)$$is a closed set ,and if $$fin V$$,$$sup_{Omega}fleq 0$$,then $$int_0^1fleq 0$$.Prove that there exists $$x_1,cdots,x_Nin Omega$$,and $$lambda_1,cdots,lambda_Ngeq 0$$,so that
begin{align*} int_0^1g=sum_{j=1}^Nlambda_jg(x_j ),forall gin V end{align*}

Below is what I can thought.Try to solve this problem by induction on $$dim V$$.

When $$N=1$$,$$V=L(f_1)$$.then $$int_0^1f_1neq 0$$, Without loss of generality，we can suppose $$int_0^1f_1>0$$.Then by the property of $$Omega$$ we have
$$sup_{Omega}f_1>0.$$
So there exists $$x_1in Omega$$,such that $$f_1(x_1)>0$$.We take$$lambda_1=frac{int_0^1f_1}{f_1(x_1)}$$,then $$lambda_1>0$$,and
$$int_0^1f_1=lambda_1f_1(x_1).$$
So the conclusion holds for $$N=1$$.