EDB — 10Q

Exercises

1. [10Q] Prerequisites:[10M]↺↻.Given \(p∈[1,∞]\) show the Minkowski inequality

   \begin{equation} \| x+y\| _ p≤ \| x\| _ p+\| y\| _ p\label{eq:dis_ Minkowski}\quad . \end{equation}

   20

   There follows that \(\| x\| _ p\) are norms.

   For \(p∈ (1,∞)\) find a simple condition (necessary and sufficient) that involves equality; compare it with [0ZY]↺↻; deduce that \(ℝ^ n\), with the norm \(\| ⋅\| _ p\) for \(p∈ (1,∞)\), is a strictly convex normed space (see [0ZZ]↺↻).

   Solution 1

   [10R]↺↻