Timeline for Orthogonality in non-inner product spaces
Current License: CC BY-SA 3.0
7 events
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| Mar 4, 2012 at 10:04 | comment | added | Ralph | @Uday: Yes, there are more notions of orthogonality in normed spaces - see the survey article quoted by Valerio. In principle you can take any norm-expression that is equivalent to orthogonality in an inner product space and use it as definition of orthogonality in a normed space. | |
| Mar 4, 2012 at 10:00 | comment | added | Ralph | @Martin: (i) $\perp_1$ is not symmetric: From above $(1,1) \perp_1 (2,0)$, but not $(2,0) \perp_1 (1,1)$ (take $t = -1$). (ii) $\perp_1$ is invariant under scalar multiplication, as is aparant from the definition. $\perp_2$ ist not: Again from above, $(0,1) \perp_2 (2,1)$ but not $3 \cdot (0,1) \perp_2 (2,1)$. | |
| Mar 4, 2012 at 2:16 | comment | added | Uday | Another question: Are there any more non-equivalent orthogonality definitions? | |
| Mar 4, 2012 at 2:07 | comment | added | Uday | Another follow up question I have is: Since $\perp_{1}$ and $\perp_{2}$ are equivalent in inner-product spaces, in what way normed spaces are deficient so that these orthogonality concepts don't agree? | |
| Mar 4, 2012 at 1:59 | vote | accept | Uday | ||
| Mar 4, 2012 at 1:43 | comment | added | shuhalo | Follow up questions: (i) Are these two notions of orthogonality symmetric? $\perp_2$ certainly is, but what about $\perp_1$? (ii) Are these notions invariant under scaling? For example, if $x \perp_1 y$, then $x \perp_1 \beta y$ for any $\beta \in K$. Most importantly, (iii): In which contexts are these two notions of orthogonality useful? | |
| Mar 3, 2012 at 19:55 | history | answered | Ralph | CC BY-SA 3.0 |