On space $X=C([0,1])$ we are given an operator $T:X \to X$ with $$ Tf(x)=\int_0^x{f(y)}dy$$ I proved using Arzela-Ascoli that the operator is compact.
The other question is to prove that $T(B_X)$ is not closed (although its closure is compact) where $B_X$ is closed unit ball in $X$. I was trying to construct a sequence $(f_n) \in B_X$ where $Tf_n \to f$ and $f$ is not in $T(B_X)$ but I failed. I tried with some polynomials, trigonometric functions, exponential, but no success. Any idea how to construct the sequence or on some other way prove the above fact.