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Dec 18 |
awarded | Notable Question |
Dec 17 |
awarded | fa.functional-analysis |
Dec 16 |
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Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
@NarutakaOZAWA: Can you explain the "by Hahn--Banach" part in more detail? I'm worried because I don't see where you used the completeness of $X$. |
Dec 16 |
revised |
Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
added 2 characters in body |
Dec 15 |
answered | Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem? |
Dec 12 |
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Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
See also math.stackexchange.com/questions/146910/…. |
Dec 12 |
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Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
You could see Theorem 27.26 of Schechter's Handbook of Analysis and its Foundations (google books) where it is proved that every space that satisfies UBT also satisfies CGT. I expect one could extract a direct proof from this. I don't have a copy of the book handy right now but I will look when I have a chance. |
Dec 12 |
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Speed of Approach to Invariant Measure
@DelioMugnolo: "Kernel is the constants" and "spectrum is discrete" are two different things to show. |
Dec 12 |
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Speed of Approach to Invariant Measure
Of course you do have $\beta = 0$ since $T_t 1 = 1$. You do not expect $T_t f$ to converge to $0$ but to $\int f\,d\mu$. In a nice symmetric case like Ornstein-Uhlenbeck you can use a spectral gap: if $N$ is the Ornstein-Uhlenbeck generator on $L^2(\mu)$, you can show its kernel is the constants and its spectrum is discrete. So if $\lambda$ is the least positive eigenvalue, by the spectral theorem you get $\|T_t f - \int f\,d\mu\|_{L^2(\mu)} \le e^{-\lambda t} \|f - \int f\,d\mu\|_{L^2(\mu)}$, which shows exponentially fast convergence. |
Dec 10 |
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Is the on-diagonal heat kernel “local” with respect to the metric?
As an easy way to see this cannot be true, consider $X$ compact and connected. For fixed $z$, as $t \to \infty$ we have $K_A(t; z,y) \to 1/\mathrm{Vol}_A(X)$ uniformly in $y$. (This is because $K_A(t;z,\cdot)$ must converge to a constant, and it also must integrate to 1 with respect to $d\mathrm{Vol}_A$.) Now let $\mu_B$ be any metric which agrees with $\mu_A$ on $U$ but gives different total volume to $X$. For sufficiently large $t$, $K_A(t;z, y)$ and $K_B(t;z,y)$ must differ for all $y$. |
Dec 3 |
revised |
Inverse problem to solve out current in the radiation problem
format |
Dec 2 |
revised |
Square roots of the Laplace operator
improve formatting, since question was already bumped |
Dec 2 |
revised |
Square roots of the Laplace operator
fix formatting |
Dec 2 |
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The name of the group
Are you sure this is a group? I don't think 1 has an inverse. |
Nov 28 |
awarded | Good Answer |
Nov 24 |
answered | Examples of common false beliefs in mathematics |
Nov 24 |
awarded | Electorate |
Nov 22 |
revised |
How did Bernoulli prove L'Hôpital's rule?
remove tag-removed, add more useful tags |
Nov 22 |
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What is the best reference for Spectral theory?
Dunford and Schwartz is of course the Holy Bible of All Things Functional Analysis, but I think it would be really heavy going for a beginner. |
Nov 22 |
answered | What is the best reference for Spectral theory? |