John

Hey guys/girls, I am a having a little problem with big/small O notation. Say that $r_n\to \infty$ and $q_n \to 0$ what is the O/o relationship between these sequences if $r_nq_n \to 0$?

Hi there :) I have a quick question that you might easily help me with. Suppose that f(X,Y) is independent of Y, then there exists a function g such that f(X,Y)=g(X) almost surely. Here X and Y are random variables. How do I see this?

@MikeMiller Thanks for the sanity check, I think its time to go to bed soon :)

@MikeMiller Sure, we can't say much "new". But I was thinking about different ways to represent this. I have come to the conclusion that any column of $A$ can be written as $Bx$ for some $x$. I am not entirely sure, do you agree?

Hi there, I have a problem that I think is rather simple. Consider to matrices A and B. If i know that $R(A) \subseteq R(B)$, what can I say about the matrices? Here $R(A)$ is the range/image/span of $A$.

@Secret Yes I think i know what you are suggesting, I will explore this. Thanks

Sorry was mean to read: If R(Z)^\perp \cap R(PZ)^\perp contains a non-zero element then R(Z) \subset R(A).

@Secret Maybe it help with the concrete problem I'm working on. Let P we a orthogonal projection onto R(A), where R(A) and R(Z) denote the range of a linear transformations A and Z onto R^n. If R(Z)^\perp \cap R(PZ)P\perp contains a non-zero element, then R(Z) \subset R(P). This is at least what is claimed in an article im reading, but I can't see why this holds.

@Secret what if we add that $V\not \perp W$

@Secret you are right, my intuition was all wrong. thanks

Hi there, can anybody help me with this seemingly easy (intuitively at least) fact: Let $V,W$ be two subspaces of $\mathbb{R^n}$. If there exists a $x\in \mathbb{R}^n \setminus\{0\}$ such that $
x\in V\cap W$, then it must hold that either $V\subseteq W$ or $W\subseteq V$.

@TedShifrin Thanks.

Quick question: I had a bounty question, and on the last day there came an answer. Now after the bounty has expired (I have been unavailable) I can't award the bounty to the answer. How com this is?

Does anyone have any references for multidimensional taylor expansions, in that the functions take values in R^n for n>1 ?

@AkivaWeinberger In regards to my question, is it a no-go to write such a statement in a thesis?

Zup,

Hey guys, can one of you answer this quick question: Consider a Hilbert space $H$ and let its continuous dual space be $H^*$. If $h^*(a)=h^*(b)$ for all $h^*\in H^*$ does it hold that $a=b$?

Hey guys, do any of you know a good reference for the space $l^2(\mathbb{N})$ that contains some kind of exploration of the Borel Sigma algebra on it?

i.e. the trivial sigma algebra?

Hey guys, how do i realize that every metric space consiting of only a singleton $\{x\}$ then the induced Borel sigma algebra is always given by $\{\emptyset, \{x\} \}$ ?

@GFauxPas Thanks mate, thats exactly what im looking for. Already found a reference :)

i.e. that $|f(x,y)|^2\leq f(x,x)f(y,y)$-

Is there a name for a mapping $f:X\times X\to \mathbb{R}$ which satisfies conditions of inner product except $f(x,x)=0 \iff x=$. Im looking for a reference that states that the Cauchy-Schwarz inequality also holds for such mappings.

Never mind im clearly a newb.

*every finite measure.

Hey guys, Do any of you know if every measure, can be written as a linear combination of probability measures?

@MikeMiller Thanks mate :)

@te

@TedShifrin Ahh i see, yeah i can see that if the hilbert space in question has more that two elements in its basis, then no dual element is injective. Thanks.

Well consider the identity mapping from R to R, that is surely a injective bounded linear map in the dual of R right?

@MikeMiller was that a response to my question?

Quick question I hope one of you can help me with: Does there exists a separable Hilbert space with a continuous dual space consisting of purely non-injective mappings?

Is it well-defined to talk about a "linear map" from a real vector space to a complex vector space. For example the map $\mu\mapsto \int f d\mu$, with suitable real vector space of measures and a complex function $f$. I would like to conclude that the above mapping is injective on some subspace, and to prove that I would like to use that linear mappings are injective iff kernel only contains zero.

@PhysicsGuy Sure, thats why i wont take your word for it, and try to prove it instead :)

@Phy

@PhysicsGuy Okay thanks, ill try to prove it formally then :)

Hey guys I have a quick question: Let $(X,d)$ be a metric space and let $(\tilde(X),d)$ be its completion. There exists an isometric embedding $I:X\to\tilde(X)$, but does it holds that the closure of $I(X)$ is $\tilde(X)$, where the closure is with respect to the topology on $\tilde(X)$?

I got this unanswered question with at bounty of 150 that ends in 5 hours. Any answer which even just has a useful reference will be awarded the bounty. math.stackexchange.com/questions/2035989/…

Hey guys - Hope you can answer a quick question. Does it make sense to consider the tensor product of two Hilbert spaces with different scalar fields? Most Litterature considers both fields either real or complex, but as you can see in my latest question i seem to extend the procedure to the mixed case.

@ColleenV Great I will give it a read. Thanks again :)

@ColleenV Okay, I will try to find it. Thank you very much.

Do you have some reference for these rules? Every webpage I read about whether to use plural or singular verbs only considers two subjects joined by either "and" or "or".

Yes, (A1) to (An) are separately defined elsewhere. So "(A1) to (An)" becomes singular - interesting. What about "(A1) to (An) and (B1)". Does this then become plural?

Can anybody help me with this grammar question: Say that I have some assumptions (A1) to (An). Is the subject (A1) to (An) singular or plural? E.g. is it "Assume that (A1) to (An) hold/holds" which is correct?

Hi Mods, I had a bounty question, and on the last day there came an answer. I have been unavailable, so i failed to award the bounty in the active period it seems and now it is expired I think. I think it is unreasonable that the answer gets no bounty, due to me not being available. Can we award the bounty to the answer somehow?

@TheGreatDuck math.stackexchange.com/questions/2035989/… Okay fair enough, just seems like a waste of 150 point.

I have an unaswered question, with a bounty that just expired. May a friend of mine post an empty answer and collect the bounty so it is'nt completely wasted?