In $\mathbb{R}^n$ the three norms $\|\cdot\|_1$,$\|\cdot\|_2$ and $\|\cdot\|_{\infty}$ verify that for any vector $v \in \mathbb{R}^n$ such that $v=\sum a_ie_i$, where the $e_i$'s are the standard basis vectors, it must be: $$|a_i|\leq\|v\|_j$$ where $i=1,\ldots ,n$ and $j=1,2,\infty$.
So, I wonder if it is the case that for any norm $\|\cdot\|$ in a (possibly finite dimensional) vector space $V$ it must hold that for any vector $v \in V$ s. t. $v=\sum a_ie_i$, where the $e_i$'s are the vectors of a normalized basis, the inequality above also holds.
I couldn't prove it by simply using the definition of the norm, so maybe there are more hypothesis needed to make the claim true. Any thoughts on how to do it?
Bessel's inequality gives you something of that flavor. More or less it says that if you have an orthonormal basis (finite or countable) of a Hilbert space, then the sum of the squares of the moduli of the coefficients is less than or equal the square of the norm of that element, i.e.
$$\sum_{n=1}^{\infty} | \langle v, e_i \rangle |^2=\sum_{n=1}^{\infty} |a_i|^2 \leq \|v\|^2$$