Let $X$ be an infinite-dimensional normed space (can assume Banach if needed). Given an integer $n\in\mathbb{N}$ and a mapping $f:\{1,\dots,n\}^2\to\{\pm 1\}$, I want to construct $\{x_1,\dots,x_n\}\subset X$ and $\{x_1^*,\dots,x_n^*\}\subset X^*$ ($X^*$ is the topological dual) such that the following are true:
- $x_i^*(x_j)=f(i,j)$,
- $\max(\|x_i\|,\|x_i^*\|)\leq C$ for all $i=\{1,\dots,n\}$ and $C>0$ independent of $n$.
The second condition makes it somewhat hard, since otherwise one could simply choose any linearly independent $x_1,\dots,x_n$ and decalare $x^*_i(x_j)=f(i,j)$.
This is true in the simple cases $X=\ell^1$ or $X=\ell^\infty$. In fact, define $x_i(j)=\delta_{i,j}$ and $x_i^*=(f_{i,1},\dots,f_{i,n},0,\dots)$, but this doesn't work for $X=\ell^p$ for $p\in(1,\infty)$ for example.