Existence of Fredholm operators between two Banach spaces

Let Ω be a nonempty open subset of R. A consequence of (i) is that there is no Fredholm operator between L(Ω) and L(Ω), 1 < p < ∞. A consequence of (ii) is that there is no Fredholm operator between L(Ω) and L∞(Ω). Proof. Put X0 = kerT and Y1 = T (X). Then dimX0 < ∞, codimY1 < ∞, and Y1 is closed in Y . Since X0 is finite dimensional, it has an algebraic topological complement X1. Then T |X1 : X1 → Y1 is an isomorphism (in the category of linear continuous maps). Since Y1 is closed and has finite codimension in Y , it has an algebraic topological complement Y0. Then dimY0 <∞ and Y0 is closed in Y . We have