Exercises
[0ZZ] Prerequisites:[106], [19D], [0ZX].Difficulty:*.We will say that the normed space \((X,\| ⋅\| )\) is strictly convex 1 if the following equivalent properties apply.
The disc \(D=\{ x∈ X:\| x\| ≤ 1\} \) is strictly convex. 2
The sphere \(\{ x∈ X,\| x\| =1\} \) does not contain non-trivial segments (that is, segments of positive length).
For \(v,w∈ D\) with \(\| v\| =\| w\| =1\) and \(v≠ w\), for every \(t\) such that \(0{\lt}t{\lt}1\), we have that \(\| t v+(1-t)w\| {\lt}1\).
For every \(v,w∈ X\) that are linearly independent we have \( \| v+w\| {\lt} \| v\| +\| w\| \quad .\)
Show that the previous four clauses are equivalent.
[ [101]]
Solution 1