Let $D: X\to Y$ and $L: Z\to Y$ are bounded linear operators between Banach spaces, and we further assume $D$ is Fredholm. Then can we conclude that the operator $$ D\oplus L: X\oplus Z \to Y$$ has a closed image?
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the image must contain that of $D.$ $D$'s image is of finite-codimension. Subspaces of finite co-dimension are closed so yes.
(to prove the last consider a mapping from the subspace plus a finite dim v-space that is bijective to the target: id on the subspace and mapping to a basis on the finite dim part.)
Mark Joshi
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Thank you. I sort of understand your answer. But, would you mind making your argument more precise? – Hang Mar 30 '17 at 21:47
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which bit do you want more clarification on? – Mark Joshi Mar 30 '17 at 22:36
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http://math.stackexchange.com/questions/2670/is-the-closedness-of-the-image-of-a-fredholm-operator-implied-by-the-finiteness?rq=1 – Mark Joshi Mar 30 '17 at 23:12