Paper 2, Section II, F

(a) Let $(X, d)$ be a metric space, $A$ a non-empty subset of $X$ and $f: A \rightarrow \mathbb{R}$. Define what it means for $f$ to be Lipschitz. If $f$ is Lipschitz with Lipschitz constant $L$ and if

$F(x)=\inf _{y \in A}(f(y)+L d(x, y))$

for each $x \in X$, show that $F(x)=f(x)$ for each $x \in A$ and that $F: X \rightarrow \mathbb{R}$ is Lipschitz with Lipschitz constant $L$. (Be sure to justify that $F(x) \in \mathbb{R}$, i.e. that the infimum is finite for every $x \in X$.)

(b) What does it mean to say that two norms on a vector space are Lipschitz equivalent?

Let $V$ be an $n$-dimensional real vector space equipped with a norm $\|$. Let $\left\{e_{1}, e_{2}, \ldots, e_{n}\right\}$ be a basis for $V$. Show that the map $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ defined by $g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left\|x_{1} e_{1}+x_{2} e_{2}+\ldots+x_{n} e_{n}\right\|$ is continuous. Deduce that any two norms on $V$ are Lipschitz equivalent.

(c) Prove that for each positive integer $n$ and each $a \in(0,1]$, there is a constant $C>0$ with the following property: for every polynomial $p$ of degree $\leqslant n$, there is a point $y \in[0, a]$ such that

$\sup _{x \in[0,1]}\left|p^{\prime}(x)\right| \leqslant C|p(y)|$

where $p^{\prime}$ is the derivative of $p$.

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