# $T$ is bounded linear operator

Question: Let $$T:C[0,1] \to C[0,1]$$, where $$T(f) = (t^{2} +2) f(t)$$. Then $$T$$ is bounded linear operator.

Is this a true or a false statement? Justify your answer.

My solution: $$| T(f) | = | (t^{2} +2) f(t) | \leq \|(t^{2} +2)\| \|f(t)\| \leq 3 \|f\|$$ which is finite since it's bounded

Is it true?

Thanks a lot.

• Note that boundedness is not the only condition you have to check, also linearity ofcourse. – Bo5man Dec 19 '18 at 13:37

## 1 Answer

I think that $$C[0,1]$$ is equipped with the norm $$||f||:= \max \{|f(t)|: t \in [0,1]\}.$$

Your solution is correct, but not nicely written.

For $$t \in [0,1]$$ and $$f \in C[0,1]$$ we have

$$|T(f)(t)|=(t^2+2)|f(t)| \le 3 |f(t)| \le 3 ||f||.$$

Thus

$$||T(f)|| \le 3 ||f||$$, which shows that $$T$$ is bounded.