about proof of linear operators

Question

$T\colon H^1[0,1]\to L^2[0,1]$ $$ T_{x}(t)=x(t) $$ $H^1[a,b]$ is the space of all continuous differentiable functions with the norm $$ \|x\|_{H^1}=\left(\int_{a}^{b}x^2(t)dt+\int_{a}^{b}((x′(t))^2dt \right)^{(1/2)} $$ firstly we need to show $T$ is linear and bounded and then how can we find its norm?

Answer 1

For any $x\in H^1([0,1])$, $T(x)=x$ thus $T$ is obviously linear.

To prove that $T$ is bounded, you just have to show that $$\|x\|_{L_2}=\|T(x)\|_{L^2}\leq C\|x\|_{H^1}.$$ And an other way to write $\|x\|_{H^1}$ is $$\|x\|_{H^1}=\sqrt{\|x\|_{L^2}^2+\|x'\|_{L^2}^2}$$ so that you can easily find the constant $C$ you need.