Understanding why linear independence leads to $f$ takes positive values .

Question

Here is Theorem 4 and its proof (from Royden 4th edition "Real Analysis"):

But I do not understand why in the second part in finding $c_2,$ in the second paragraph from below: $\{e_{1}, ... , e_{n}\}$ being linearly independent leads to that $f$ takes positive values on the boundary of the unit ball, could anyone explain this for me ,please?

Answer 1

Basically linear independence here guarantees that $\sum_{i=1}^nx_ie_i$ is not $0$.

By definition of norm, $||\sum_{i=1}^n x_ie_i||=0$ iff $\sum_{i=1}^n x_ie_i=0$, and by linear independence, that happens only if $x_i=0$ for all $i$. Since $x$ is on the unit ball, this cannot happen, so $\sum_{i=1}^n x_ie_i\neq 0$, and $||\sum_{i=1}^n x_ie_i||>0$.