what is an example of a normed space such that $||\sum x_i||=\sum ||x_i||$ does not imply they have the same direction?

Question

Let $V$ be a normed space over $\mathbb{K}$ and $x_1,...,x_n\in V\setminus\{0\}$ such that $||\sum_{i=1}^n x_i||=\sum_{i=1}^n ||x_i||$.

If $||\cdot||$ satisfies parallel law, then this implies that $x_i=c_ix_1$ for each $i$ where $c_i$ is a positive real.

However, I doubt this holds in general.

what is an example of a normed space such that $x_i\neq c_ix_1$ for some $i$?

Answer 1

Take $V=L^\infty[0,1]$, $f_1(x)=1$ for all $x$, and $f_2(x)=0$ for $x \in[0,1/2]$, and $f_2(x)=1$ for $x \in(1/2,1]$.

Another example:

$V=C[0,1]$, $f_1 \equiv 1$ and $f_2(x)=x$, the norm being sup-norm.