Examples of Problems with Solutions

A functional $f$ on the subspace $G = \left\{ \left(x_{1}, 0\right) \mid x_{1} \in \mathbb {R}\right\}$ of the space $\mathbb {R}^{2}$ is defined by the formula $f(x) = \alpha x_{1}, x = \left(x_{1}, 0\right) \in G$, where $\alpha \in \mathbb {R}$ is fixed.

Describe all linear extensions $F$ of the functional $f$ to $\mathbb {R}^{2}$. For which of these extensions $\left\| F\right\| = \left\| f\right\|$, if $\mathbb {R}^{2}$ is endowed with the norm $\left\| \cdot \right\|_{p}, 1 \leq p \leq +\infty$? In which cases the extension that preserves the norm is unique?

Let $G = \left\{ x \in C(\left[0,1\right]) \mid \int_{0}^{1} x(t) d t = 0, x(1) = 0\right\}$. Construct a functional $f \in (C(\left[0,1\right]))^{*}$ such that $f(x) = 0, x \in G$, and $f\left(x_{1}\right) = f\left(x_{2}\right) = 1$, where $x_{1}(t) = 1, x_{2}(t) = 1-t, t \in [0,1]$.

Prove that there exists a nonzero functional $F \in \left(L_{\infty }(\left[a, b\right])\right)^{*}$ such that