Discussion Board Lecture 02
small comment on Theorem 2.2 (c)
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Dear ISemTeam, in order to avoid boring case differentiation it might be convenient to make explicitely the general agreement that whenever necessary it will be assumed that the considered normed spaces are non-trivial. Otherwise one has to explain, e.g., in Theorem 2.2 (c) how $ B(\lambda,\frac{1}{\|R(\lambda,A)\|}) $ has to be read in the case that $ X=\{0\} $. Best wishes, Heiko
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Dear Heiko, thanks for pointing this out; we had not thought of this problem. However, the intention to exclude the trivial Banach space by definition does not seem happy to me. For me, the solution lies in admitting balls with radius $ \infty $. When writing the lectures I always had in mind than one should at some point disclose the meaning of $ B(x_0,r) $, but I never discovered a good place for it. So, please think directly before Theorem 2.2 something like the following definition. For a metric space $ (M,d) $, a point $ x\in M $, and $ r\in(0,\infty] $, we define the open and closed balls with centre $ x $ and radius $ r $, $ B(x,r):=\{y\in M;\ d(x,y)<r\},\qquad B[x,r]:=\{y\in M;\ d(x,y)\leqslant r\} $, respectively. Then, accepting $ \frac10=\infty $, one also gets the asserted statement. Best wishes, Jürgen
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Dear Heiko, one could just exclude $ X=\{0\} $ as a matter of principle, but I'd also prefer not to do that. For example, when one considers the product space $ X\times Y $, then sometimes it is useful to consider the case $ Y=\{0\} $. I don't claim that for this it's necessary to consider $ Y=\{0\} $ as a Banach space, but I consider it helpful. Best wishes, Hendrik
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Missing assumption in Lemma 2.10
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Dear ISem team, in Lemma 2.10 the assumption is missing that $ dom(A) $ is dense (which is used in the proof of part (a) and implicitly also in the proof of part (b)). Best wishes, Johannes
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Dear Johannes, yes! Thanks for finding the omission. Best wishes, Jürgen
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Comment on notation in 2.13 Remarks (a) and Proof of Theorem 2.12
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Dear Isem-Team, I noticed that you mixed up your notation in Remark 2.13 (a) and in the second line of the proof of Theorem 2.12: I suppose you wanted to write $ \left(I-\frac{t}{n}A \right)^{-1}=\frac{n}{t}\left(\frac{n}{t}I-A \right)^{-1} $ in the remark and $ \left(-\frac{n^2}{t^3}I+\frac{n}{t^2}A+\frac{n^2}{t^3}I\right)\ldots $ in the proof. So there is only a missing identity. Best wishes, Julia
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Dear Julia, thanks for pointing this out. You are quite correct; it is only later that we want to introduce (and use) this abbreviating notation. Best wishes, Jürgen
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Note on notation in Theorem 2.2
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For $ z_0\in\mathbb{K} $ and $ r>0 $ we write $ B(z_0,r) := \{z\in\mathbb{K};\; |z-z_0|<r\} $ for the open ball of radius $ r $ around $ z_0 $.
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