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4h
comment Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?
In a more sophisticated sense, the answer has to be "yes" because it certainly isn't Banach, and you didn't use the axiom of choice. If I've done my work correctly, in a model of ZF+DC in which every set of reals has the Baire property, we can show that every incomplete separable normed space is meager. (As a subset of its completion, it has the BP, so it is meager in its completion, and an elementary argument shows it must therefore be meager in itself.)
4h
comment Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?
@Ian: In some sense they are actually equivalent. You can show that if $X$ is a topological space and $A$ is a dense subset of $X$ then $A$ is meager in itself (in the subspace topology) iff it is meager in $X$.
8h
answered Equality in definition of dual space norm
11h
answered Is the inequality $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ true?
11h
comment Weak convergence in the Sobolev space and compact embeddedness
Your last sentence seems to be truncated.
11h
comment Weak convergence in the Sobolev space and compact embeddedness
Just to repeat a point from below, your step 1 is false as written - you can only conclude that there is a subsequence which converges weakly in $H$. But that is enough to run the argument.
19h
answered Conditional Expectation: Sum inside or outside?
19h
answered Predicate logic inference in a simple proof of uniform continuity.
20h
revised Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?
added 1773 characters in body
20h
comment Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?
I'm a working probabilist, and I have to tell you that the ordered triple notation $(\Omega, \mathcal{F}, P)$ is absolutely universal in this field - anything else looks weird. I'm sorry you don't like it!
20h
comment Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?
@Potato: It's quite common in probability books/papers to "suppress" the underlying probability space. That's what's happening here: the pushforward measure depends on $X$ and also implicitly on the underlying measure $P$, but we often omit the underlying measure from our notation. For instance, every time we say "almost surely", we are implicitly using the measure $P$, but we don't say so unless it's necessary to avoid ambiguity.
20h
comment Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?
I understand what you mean, but I agree with @Potato that it is nonstandard notation, and looks weird.
20h
answered Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?
1d
answered Example of equicontinuous sequence of functions which is not convergent
1d
comment Radon- Nikodym Theorem For von Neumann algebras
A better way to use italics on this site is to enclose the word in * symbols, instead of using math mode (which looks strange and renders slowly). But are you sure you need to use italics as much as you do? When almost every other word is emphasized, it really loses its impact.
1d
comment High computation in probability
When you quote a problem, please give the source. This appears to be from the 2011 AIME.
1d
comment Deny Lions Lemma
Never mind, I found it on Google Books.
1d
answered Deny Lions Lemma
1d
comment Deny Lions Lemma
For those who do not have the book, can you explain some more about how the linear functionals $f_i$ are chosen, and in particular, how many of them there are? And also, what exactly is $P_k$?
1d
comment Show that for any random variable $X$, and any $a > 0$, $P(|X| > a) \leq {EX^4 \over a^4}$.
Hint: $|X| > a$ iff $|X|^4 > a^4$.