Banach Space Problem: Proving Subspaces Contain e-Orthogonal Elements

[Carl140]

Homework Statement

Let E be a Banach space and let M be a closed subspace of E. A
vector x in E is called e-orthogonal to M if for all y in M the following inequality holds: ||x+y||>= (1- e)||x||.


Prove that for each e>0 any proper subspace of M contains e-orthogonal
elements.

The Attempt at a Solution

Well clearly 0 is always such an element but how to find more elements?

 

[morphism]

What is e? A positive number less than 1? And did you really mean to say that M contains e-orthogonal elements? Because the only such possible element is 0. Proof: if x is in M and x is e-orthogonal to M, then ||x||(1-e) <= ||x+(-x)||=0.

Edit: And if you meant to say "E contains elements which are e-orthogonal to M", then this result is known as (or at least immediately follows from) the Riesz Lemma.