From: Stephen Montgomery-Smith Subject: Re: Basis in Banach spaces Date: 6 Feb 2000 23:00:01 -0600 Newsgroups: sci.math.research Summary: [missing] Enrico Talinucci wrote: > > > B is an n-dimensional Banach space on the field K, with norm ||.|| . > b_1 is a given nonzero vector in B. > I need to know if there exists a basis of B containing b_1, let's say > (b_1,...,b_n), with the following property: > > For any v \in B: > Let v_1,...,v_n be the vector components of v with respect to the basis > (b); i.e. v may be written as > v = v_1 + ... + v_n > where v_1 = A_1 b_1, ..., v_n = A_n b_n, with A_1,...,A_n \in K. > Then: > ||v_1|| < ||v|| (or =) If I am getting your question correctly, I think what you want comes from the Hahn-Banach Theorem. Given b1, let B be the 1-dim vector space generated by b1. Let p:B->K be the map p(t b1) = t ||b1||. Clearly p has norm 1. Use the Hahn Banach Theorem to extend this to a map P:B->K, also with norm 1. Let b2,...,bn be any basis of the kernel of P. Now suppose v = v_1 + ... + v_n as above. Then P(v) = A_1 ||b1||, that is, |P(v)| = ||v1||. Since P has norm 1, you get what you desire. -- Stephen Montgomery-Smith stephen@math.missouri.edu 307 Math Science Building stephen@showme.missouri.edu Department of Mathematics stephen@missouri.edu University of Missouri-Columbia Columbia, MO 65211 USA Phone (573) 882 4540 Fax (573) 882 1869 http://www.math.missouri.edu/~stephen