A thought just occurred to me, in a Banach space $X$, what is $$\lVert c \rVert_{X}=c\lVert \text{Id} \rVert_{X}$$ where $c$ is a constant? Is it even defined?
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It's not really clear what you're asking here. Is $c$ the linear map $X \to X$ that scales everything by $c$? Then its norm is $c$. – Najib Idrissi May 17 '13 at 17:10
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@nik No $c \in \mathbb{R}$. It is in the field of the Banach space. – matt.w May 17 '13 at 17:10
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There is a natural continuous injection from the field $k$ into the Banach space $X$ by identifying $c$ and $1.c$. – user40276 May 17 '13 at 17:11
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@user40276 But what is $1$ in a Banach space?? – matt.w May 17 '13 at 17:12
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You assumed $Id$ in the Banach space, so in your notation $1 = Id$ – user40276 May 17 '13 at 17:14
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@user40276 But does it always exist? I understand that the element $0$ exists cause it's a vector space. $0$ is the identity element under addition. But I am not sure that there is an identity element under multiplication if you know what I mean. – matt.w May 17 '13 at 17:15
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No. To use multiplication properly you must have a Banach algebra and even in this case 1 could not exist – user40276 May 17 '13 at 17:17
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@user40276 Right, so it is not defined. – matt.w May 17 '13 at 17:18
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Where did you run into this notation? Some context might help figuring out what is meant. – Martin May 17 '13 at 19:02