Fredholm index and spectral flow

for all k. This shows that the sequence of functions is uniformly equicontinuous. Let now t ∈ R and assume that the sequence (|fk(t)|) is unbounded. Then it follows from the estimate that the sequence (min[t−1,t+1] |fk|) is unbounded and therefore also that (‖fk‖L2) is unbounded, contradiction. We conclude the the sequence ‖fk(t)‖ is bounded, for every t ∈ R. Let now T > 0. Then it follows by Ascoli’s theorem that the sequence has a subsequence fkj which converges to a continuous function g ∈ C[−T, T ], uniformly on [−T, T ]. It follows that also fkj → g in L2([−T, T ]), so that g = f on [−T, T ]. This establishes the continuity of f. The final statement follows by exactly the same argument.