In the vector space spanned by $\displaystyle {\bf{e}}_1,{\bf{e}}_2,...,{\bf{e}}_n$ which are linearly independent (not necessarily the standard basis), $\displaystyle {\bf{y}}_1=\sum_{i=1}^na_i{\bf{e}}_i$ and $\displaystyle {\bf{y}}_2=\sum_{i=1}^nb_i{\bf{e}}_i$. My question is: if the distance of $\displaystyle {\bf{y}}_1$ and $\displaystyle {\bf{y}}_2$ can be arbitrarily small, can the distance of their combination coefficients $\displaystyle a_i$ and $\displaystyle b_i$ be arbitrarily small too? Assume that we are discussing in the Euclidean spaceR$\displaystyle ^n$. Thanks.