Note that saying that R(A) is closed in a Banach space is the same as saying that R(A) itself is a Banach space. Thus the earlier theorems about surjective maps apply. [Picture] Next we discussed Theorem 3.12, first for bounded operators. For closed operators you then get it by taking the graph norm on D(A). [Picture] But how about maps which are not injective? Just apply Theorem 3.12 to the natural modification of A as map of the quotient space X/N(A) to Y. This is for bounded A, for closed A do the same as before and consider A on D(A) with the graph norm. [Picture] The quotient space X/N(A) is again a Banach space because N(A) is closed. [Picture] The proof of completeness of X/M is based on first restricting to a subsequence of the Cauchy sequence to make the series consisting of the differences "absolutely" convergent. The corresponding vectors representing the cosets can then be chosen to be convergent and this basically does the job: [Picture] Ignore the right part below. On the left part we formulate Theorem 3.14 and show how it follows. [Picture] Back to N(A), R(A) and all that. We always have, almost by definition, that

N(A)=o(R(A') and N(A')=R(A)o

(I cannot do superscript small o) These are supplemented by Theorem 3.7 and now also Theorem 3.16: [Picture] [Picture]