Regarding proving a result in Functional analysis

Question

I am solving tutorials of a institute which I am not a student. Please help in this question - Question is - Let { $ u_1$ , ..., $ u_n$ } be a linearly independent set of elements in Normed linear space X. Then prove that there exists a positive real number c such that for every choice of scalars $ k_1$ ,..., $ k_n $ belonging to field K we have ||$k_1u_1 $ +.... + $ k_r u_ r$|| is greater than or equal to c ( | $k_1$ | +... + | $k_r$|). What I did - if sum in lhs =0 then for any positive real number c inequality holds. But how to do this for general sum.

Answer 1

The set $M$ of all $(k_1,k_2,\ldots,k_n)\in K^n$ such that $\sum \vert k_i\vert=1$ is compact and the mapping $M\ni (k_1,k_2,\ldots,k_n)\mapsto \Vert \sum k_i u_i \Vert$ is continuous. Since $\sum k_i u_i\not=0$ for all $(k_1,k_2,\ldots,k_n)\in M$ there is a positive constant $c$ such that $\Vert \sum k_i u_i\Vert\geq c$ on $M$. For arbitrary $0\not=(k_1,k_2,\ldots,k_n)\in K^n$ you may apply this result to $(l_1,l_2,\ldots,l_n)\in K^n$ with $l_i=k_i/\sum \vert k_j\vert$. For $0=(k_1,k_2,\ldots,k_n)$ the inequality becomes an equality.