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Dec
18
awarded  Notable Question
Dec
17
awarded  fa.functional-analysis
Dec
16
comment 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
comment 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
comment 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
comment Speed of Approach to Invariant Measure
@DelioMugnolo: "Kernel is the constants" and "spectrum is discrete" are two different things to show.
Dec
12
comment 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
comment 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
comment 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
comment 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?