I've been working in the following problem but I can't fill in all the details.
Consider $E=\mathcal{C}\big([0, 1],\mathbb C \big)$ endowed with the norm induced by the product $\displaystyle\left\langle f,g\right\rangle =\int_0^1 f(t)\overline{g(t)}\,dt$. Fix $\alpha\in (0, 1)$ and define $u_\alpha\,\colon E\to \mathbb C$ by $$\displaystyle u_\alpha(f):= \int_0^\alpha t^2 f(t)\,dt.$$ I must prove that $u_\alpha\in E^*$, and also have to find its norm. Furthermore, I have to explain why $E$ is not a Hilbert space using this operator.
It is easy to see that $u_\alpha$ is a continuous linear functional $E\to \mathbb C$ because $|\phi_\alpha(f)|\leq \sqrt{\dfrac{\alpha^5}{5}}\, \lVert f \rVert_E$, and thus $ \lVert u_\alpha \rVert \leq\sqrt{\dfrac{\alpha^5}{5}}$, and I conjectured that this must be its norm, but I have failed trying to prove that (it may not be its norm in the first place, but I haven't got anything better)
For the second item, it suffices to show that the Riesz-Fréchet representation theorem doesn't hold for $u_\alpha$ (I've done it already, it is not hard at all though it is not evident)
One last question, for $n\in \mathbb N$, what is the norm of the following operator: $\displaystyle \phi_\alpha(f)= \int_0^\alpha t^n f(t)\,dt $?
Can you help me to find such norms? Thanks in advance!
The Cauchy-Schwarz inequality becomes equality iff one of the vectors is a multiple of the other, so take $f(t) = \sqrt{\frac{5}{\alpha^5}} t^2$. Then note that $\|f \|_2 = 1$ and $$\lvert u_\alpha(f) \rvert= \sqrt{\frac{5}{\alpha^5}}\int^\alpha_0 t^4 dt = \sqrt{\frac{\alpha^5}{5}}, $$ thus the equality is achieved and so $\|u_\alpha\| = \sqrt{\frac{\alpha^5}{5}}$.
The exact same reasoning works for $$\phi_\alpha(f) = \int^\alpha_0 t^n f(t) dt;$$ just use Cauchy-Schwarz and then choose $f$ so as to attain equality.