Exercise 8, § 1

Topological Vector Spaces, V, p. 62

Let E be a real or complex normed space, having the following property : if $x, y, x', y'$ are four vectors in E such that

\|x\| = \|x'\|,\quad \|y\| = \|y'\|,\quad \|x + y\| = \|x' + y'\|,

then

\|x - y\| = \|x' - y'\|

. Show that the norm on E is prehilbertian (use exerc. 7).

Solution

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