$(a)$ Let $\mathbb{X}$ a normed space and $f \in Z^{\prime},$ where $Z$ is subspace of $\mathbb{X}.$ If $f$ have two distinct extensions $f_{1},f_{2}\in\mathbb{X}^{\prime}$ that preserve the norm of $f,$ show that $f$ have infinity linear extensions that preserve norm of $f.$
$(b)$ Consider the space $\mathbb{R}^{2}$ with norm $\|(x, y)\|=\max \{|x|,|y|\} .$ In subspace $Z=\{(t, t): t \in \mathbb{R}\}$ consider the linear functional $f:Z\to\mathbb{R}$ given by $f(t,t)=t.$ Find two distinct linear extensions defined in $\mathbb{R}^{2}$ that preserve norm of $f.$
Can anybody help me? I had no idea
Best Answer
For (a), consider convex combinations.
For (b), the norm of $f$ is 1. The obvious extensions are $f(x,y)=x$ and $f(x,y)=y$. Do these work?