This question is related to one I asked earlier here. Specifically, I am trying to show that the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$ is not compact when $T: C^{2}[0,1]\mapsto C^{1}[0,1]$. To do that, I am attempting to exhibit a bounded sequence in $C^{2}[0,1]$ whose image ...

A 1-dimensional connected manifold has the property that when you remove any one of its points you get at most two connected components. Prove this and then use it to show that your set is not a manifold.

I need to solve, in $C[0,1]$, the equation $\displaystyle x(t) - \lambda \int_{0}^{\pi}(\sin t \cos s)x(s) ds = \sin t$. Adding the integral part to both sides, I obtain $x(t) = \sin t + \lambda \int_{0}^{\pi}(\sin t \cos s)x(s) ds$, which is, I believe, a Fredholm Integral Equation of the secon...