Timeline for Which norms on vectors can be consistently decomposed?
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| S Feb 24, 2018 at 9:19 | history | suggested | Ali Taghavi |
I add two tags.
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| Feb 24, 2018 at 8:35 | review | Suggested edits | |||
| S Feb 24, 2018 at 9:19 | |||||
| Feb 23, 2018 at 16:39 | vote | accept | Mateus Araújo | ||
| Feb 23, 2018 at 15:53 | history | edited | Mateus Araújo | CC BY-SA 3.0 |
added 3 characters in body
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| Feb 23, 2018 at 15:48 | answer | added | Mateus Araújo | timeline score: 0 | |
| Feb 7, 2018 at 13:40 | comment | added | Mateus Araújo | Actually, this theorem doesn't seem to imply that the $p$-norms are the only decomposable ones, because it needs to assume that for all $x,y \in \mathbb R^n$ $|x| \le |y| \Rightarrow \|x\| \le \|y\|$, and I don't think decomposability of the norm implies this property. Still, this is a very good starting point to look for a counterexample. | |
| Feb 7, 2018 at 9:05 | comment | added | Mateus Araújo | Thanks a lot, Mikhail Ostrovskii, this is exactly what I needed. I'll write up your comment as a proper answer later. | |
| Feb 7, 2018 at 8:51 | comment | added | Mateus Araújo |
LSpice, from a norm defined on a space of a fixed dimension one can always define a norm for smaller dimensions using your padding-with-zeroes strategy. I was leaving that implicit, as this was just an informal motivation for the question.
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| Feb 7, 2018 at 5:35 | comment | added | Mikhail Ostrovskii | Or just look at Theorem 1.b.7 in Lindenstrauss-Tzafriri, Classical Banach spaces, volume II | |
| Feb 7, 2018 at 1:56 | history | edited | YCor |
edited tags
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| Feb 7, 2018 at 1:34 | comment | added | Bill Johnson | Then Google bohnenblust characterization of ell_p for information. | |
| Feb 7, 2018 at 0:17 | comment | added | LSpice | Your reformulation in terms of basis vectors suggests that your equation $\lVert(a, b, c)\rVert = \lVert(\lVert(a, b)\rVert, c)\rVert$, which doesn't seem to make sense for $\lVert\cdot\rVert$ interpreted as a norm on a space of fixed dimension (since the left-hand side as written takes norms of vectors in $k^3$, and the right-hand side of vectors in $k^2$), is actually supposed to be $$\lVert(a, b, c)\rVert = \bigl\lVert(\lVert(a, b, 0)\rVert, \lVert(0, 0, c)\rVert, 0)\bigr\rVert$$ (with $n = 3$, $P_1 = \{1, 2\}$ and $P_2 = \{3\}$). Is that correct? | |
| Feb 6, 2018 at 22:50 | comment | added | Mateus Araújo | I google it. I don't see how it helps with my question. Note that I'm asking about the finite-dimensional case, and I'm not asking about equivalence in the sense of convergence. | |
| Feb 6, 2018 at 22:16 | comment | added | Bill Johnson | Google perfectly homogeneous bases. | |
| Feb 6, 2018 at 22:09 | history | edited | Mateus Araújo | CC BY-SA 3.0 |
added 21 characters in body
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| Feb 6, 2018 at 22:07 | comment | added | Mateus Araújo | Yes, I'm sorry, I meant to ask for all partitions. | |
| Feb 6, 2018 at 20:36 | comment | added | Christian Remling | With your actual definition, there are certainly other examples such as $\|(a,b,c)\| = |a|+|c| + 2|b|$, $P_1=\{ 1, 3\}$, $P_2 = \{ 2\}$. But maybe it doesn't really formalize what you actually wanted. You could also ask for this property for all partitions, and the $p$ norms satisfy that too, of course. | |
| Feb 6, 2018 at 17:26 | history | asked | Mateus Araújo | CC BY-SA 3.0 |