🚀 go-pugleaf

Hi everybody!

I'm in trouble with the following problem.


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 =)

In other words, the norm of the component of any v with respect to the given
b_1 must be less or equal than the norm of the same v. For example, that's
clearly true for an orthogonal basis in a Hilbert space, for all vectors in
the basis.

Does such a basis exists? I don't need to build it, I just need to know if
it exists.


Any help will be greatly appreciated. Thanks in advance.

Enrico Talinucci