I am currently trying to understand a proof and ran into the following problem. The proof states (everything takes place in a commutative, unital Banach-Algebra):
A linear subspace $X$ with codimension 1 and no invertible elements contains no elements near the identity.
I tried verifying this via contradiction but it didn't work.
I do not possess much knowledge about Banach algebras but I'm familiar with the basic conceptions. I'm a bit lost and any hint regarding the strategy would be greatly appreciated.
I would guess it's a reference to the following fact: if $\|e-a\|<1$ then $a$ is invertible; in fact $a^{-1} = e + (e-a) + (e-a)^2 + \dotsb$, by the usual formula for a geometric series. (The condition $\|e-a\|<1$ ensures the series converges absolutely, so it converges.) That is, if $a$ is near the identity, it is invertible; contraposing, noninvertible elements are not near the identity.