I need just a hint, not a whole solution. Problem:
Let $$T:C[0,1]\to c$$ where $c$ is the space of the convergent sequences and $$T(f)=\left(f\left(\frac{1}{n}\right)\right)$$ prove that $T$ is bounded surjection.
What I have done:
The boundedness is easy to check. For the sujectivity lets assume that $a_{n}=f(n)\in c$ so I take $(f\circ g)(n)$ where $g(n)=\frac{1}{n}$ so $$ T((f\circ g)(n))=\left(f\left(\frac{1}{n}\right)\right)$$ But the problem is that I cant show that $f\circ g$ belongs to $c$. How can I handle this problem? Is my approach true or I should change?
It's hard to tell what you are trying to do for surjectivity. What's $f$?
What you need to do is, given $a\in c$, find $f\in C[0,1]$ with $Tf=a$. Since $a$ is convergent, say $a_n\to a_0$, you may define $f(0)=a_0$, $f(1/n)=a_n$, and now you need to "fill the gaps" so that $f$ is continuous.