I have to check if those operators are bounded and if so what are their norms.
1) $\phi:C^1[0,1]\ni f > \rightarrow\int_0^{1/2}f(t)dt+f'(2/3)\in\mathbb{R}$ with norm $\|\cdot\|=\sup_t|f(t)|+\sup_t|f'(t)|$
My attempt
Its obvious that $|\int_0^{1/2}f(t)dt+f'(2/3)|\le \sup_t|f(t)|+\sup_t|f'(t)|$. So $\|\phi\|\le1$ but I cant prove it can't be $<1$.
2) $\theta:\ell^2\ni (x_n) \rightarrow \sum_{n=2011}^{\infty}2^{-n}x_{3n} + \sum_{n=2012}^{\infty}3^{-n}x_{3n+1} \in \mathbb{R}$
My attempt
I can prove it is continuous (using Cauchy-Schwarz) but can't find the norm.