Talk:Equivalence of Definitions of Norm of Linear Transformation

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Duplicate Definitions

I noticed that Definitions 1 and 3 are equivalent.

Looking at the source on Google Books I found that one of the definitions should be equality with one.

$(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H = 1}$
Definition - Norm of Bounded Linear Transformation.png


I'm not sure what the problem is.
Yes, they're equivalent (I presume), that's the point of this page. Thus we have 4 equivalent definitions (each one supposedly sourced from a source work of some kind, although that may not be the case here, I'm not sure of the provenance of these definitions) and the idea of this page is to demonstrate that they do indeed define the same object.
So if some sources give the definition like this:
$\norm A = \sup \set {\norm {A h}_K: \norm h_H = 1}$
... and some give it like this:
$\norm A = \sup \set {\norm {A h}_K: \norm h_H = 1}$
... then the proof on this page (once it has been crafted) is sufficient to justify merely invoking Definition:Norm on Bounded Linear Transformation and the author of whatever page they are writing doesn't need to worry about which definition they use in their exposition -- because they have all been proven to be equivalent, it does not matter.
What we actually do need to do here is to write each of these definitions into a separate definition page, and transclude them as necessary in the same way that we do with every other multiple definition.
If it turns out that these definitions are not actually equivalent, then we need to tear this page down and rebuild it.
Unfortunately I don't understand this area too well, so I'm not the man to do this work. --prime mover (talk) 22:43, 4 May 2021 (UTC)
I misused the English language. I wrote "equivalent" when I meant "identical"
Before I changed the page, Definitions 1 and 3 were:
$(1): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
$(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
So I changed Definition 3 to be:
$(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H = 1}$
as per the source. --Leigh.Samphier (talk) 01:20, 5 May 2021 (UTC)
Beg your pardon, I see what you mean now. I was looking at the definition page after you'd edited it and got confused.
Didn't help with the edit made by Daniel.minaya who, while correcting the definition, completely changed the notation. I rolled it back routinely as I had completely missed the fact that he had actually fixed the mistake.
As I say, it's not a page I've had much input into, so I didn't study it closely. --prime mover (talk) 05:39, 5 May 2021 (UTC)