Let $T : (C([-1,1]),||.||_{\infty}) \rightarrow (C([-1,1]),||.||_{\infty}) $
Such as : $(Tf)(x)=\frac{1}{2}(f(x)+f(-x))$ . For all $f\in C([-1,1])$
Why $T$ isn't Compact ?
I tried to use the sequence $f_n(x)=x^n$. For $x\in [-1,1]$.
But I couldn't prove that $(T(f_n))_n$ has no convergent subsequence.
Best Answer
Hint: Instead, try $$ f_n(x) = x^{2n}, $$ and observe that any subsequence of $Tf_n=f_n$ converges pointwise to a discontinuous function.