Timeline for A question about the proof of Riesz-Thorin interpolation theorem
Current License: CC BY-SA 4.0
12 events
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| Apr 6, 2019 at 8:26 | comment | added | Jochen Glueck | @aurora_borealis: Hmm... What makes you so sure that the $L^p$-spaces in the notes you linked are supposed to consist of real-valued functions rather than of complex-valued functions? As far as I can see, this is not clarified in the notes. But anyway, the norm of an operator on a real-valued $L^p$-space does indeed coincide with the norm of the canonical extension of this operator to the complex-valued $L^p$-space (see my comments below Iosif Pinelis' answer below). | |
| Apr 6, 2019 at 7:52 | comment | added | aurora_borealis | @JochenGlueck By definition $N_0$ is the operator norm between the space of real functions on $\mathbb{R}^n$, while in the proof the function $f_{yi}$ are complex in general, we can't immediately claim they are the same, as the example showed below. You can check it in matrix case. | |
| Apr 6, 2019 at 7:36 | vote | accept | aurora_borealis | ||
| Apr 5, 2019 at 15:54 | comment | added | Jochen Glueck | The formulation of the question seems to indicate somekind of misunderstanding: the space $\mathbb{R}^n$ in the linked notes is the underlying measure space of $L_p$. I can't see why the measure space $\mathbb{R^n}$ should be changed to $\mathbb{C}^n$ anywhere in the argument. | |
| Apr 5, 2019 at 13:26 | answer | added | Iosif Pinelis | timeline score: 2 | |
| Apr 5, 2019 at 5:33 | comment | added | aurora_borealis | @IosifPinelis Sorry I didn't state it clearly. I have changed my statement and listed my questions above. | |
| Apr 5, 2019 at 5:29 | history | edited | aurora_borealis | CC BY-SA 4.0 |
To show my questions explicitly.
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| Apr 4, 2019 at 20:40 | review | Close votes | |||
| Apr 9, 2019 at 9:27 | |||||
| Apr 4, 2019 at 15:03 | comment | added | Iosif Pinelis | If you don't see such a statement there (in the lecture notes?), then what is your question? | |
| Apr 4, 2019 at 12:49 | comment | added | aurora_borealis | @losifPinelis Me too, so I am a little doubt about this proof... or are there easier way to prove this theorem for the case $T$ is a symmetric matrix? | |
| Apr 4, 2019 at 12:39 | comment | added | Iosif Pinelis | I am not finding there any statement that the real and complex versions of the norm are the same. | |
| Apr 4, 2019 at 12:17 | history | asked | aurora_borealis | CC BY-SA 4.0 |