1. The problem statement, all variables and given/known data 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. 3. The attempt at a solution Well clearly 0 is always such an element but how to find more elements?