EDB — 106

Exercises

1. [106] Prerequisites:[0Q0]↺↻.

   Let again \(X\) be a normed vector space with norm \(\| ⋅\| \). Let \(B(x,r){\stackrel{.}{=}}\{ y∈ X : \| x-y\| {\lt} r\} \) be the ball. Let \(D(x,r){\stackrel{.}{=}}\{ y∈ X : \| x-y\| ≤ r\} \) be the disk. Let \(S(x,r){\stackrel{.}{=}}\{ y∈ X : \| x-y\| = r\} \) be the sphere. Show that \(\overline{B(x,r)}= D(x,r)\), that \(B(x,r)= {{D(x,r)}^\circ }\), and that \(∂{B(x,r)}=∂{D(x,r)}= S(x,r)\). Also show that \(B(x,r)\) is not closed and \(D(x,r)\) is not open.