/me extends @PedroTamaroff 's answer and adds info for @masfenix - remember norms need not be round. Take the supremum norm for example it's "circle" is a "square", this just means you can find a (assuming you use the normal pythag norm :P) you can find a ball around and inside that square, and you can put squares around a ball.
it's way too early to tell. At undergraduate level, you will have to ask the undergraduates there for their experience; in terms of faculty, both are excellent places
@Sanchez $(X,d)$ is a complete metric space, $U$ is open in $X$. Define $d'(x,y)=d(x,y)+|f(x)-f(y)|$ where $f(x)=d(x,U^c)^{-1}$. Then $(U,d')$ is complete, and $d,d'$ are topologically equivalent on $U$.
@Sanchez Well, it is quite easy now, is it not? I know $X$ is complete, I know that if $x_n$ is $d'$-Cauchy it is $d$-Cauchy, so I can find $x\in X$ such that $x_n\to x$ in $d$. Since $f$ is continuous, $x_n\to x$ in $d$ implies $x_n\to x$ in $d'$. All I have to do is prove $x\in U$ now.
Helo guys, working on sobolev spaces and there is a pretty big theorem (sobolev embedding theorem) that says if $s > t$ then $H_t \subseteq H_s$. But why is this. Intuitively, since $s$ is the larger number, by definition, all functions have upto $s$ weak derivatives. If it has $s$ weak derivatives, then it must also have $k$ derivatives..
sorry that should say $t > s$ so the space $H_t$ is embedded in $H_s$
@badass to me, "evaluate" means to compute a value. "value" is also a nebulous term; it can mean a closed expression for an integral or sum, a rational number, or a decimal approximation.
@Pedro: He doesn't state it within the next ten pages. By then, if it even exists that he states it, I would have lost all care for my problem. And so,
May you please give me an example where the root test is superior to the ratio test?
I only drew the cylinder and that's about it...I'm kind of lost at the next step, but I do know that I have to use cross product but before I do that I need to do partial derivatives
Hi guys, is there any intuitive meaning behind a compact set? If possible, I would like the intuition behind both equivalent notions: 1) X is compact if every open cover has a finite subcover. 2) X is compact if every convergent sequence in X has a convergent sub sequence
@masfenix At any rate people have asked at both math.stackexchange and mathoverflow for intuition behind compactness you can probably find these threads by searching.
@KarlKronenfeld thanks Karl. Did not know they were not equivalent in a non metric space. Although I havn't taken topology so thats why I said it above.
By the universal property, there is a bijection between the set of arrows $A\to X\times Y$ and the set of pairs of arrows $A\to X$, $A\to Y$ for fixed $A$.
In symbols there is $\varphi_A:\text{hom}(A,X\times Y)\cong F(A)$, where $F$ is a functor we will define next.
@AlexanderGruber Hm, let me think a second (I am thinking of it as an arrow in a category of slice categories but you may have a better interpretation)
@AlexanderGruber It's a composite with a hom-functor, yes.
i mean, "___ is representable" implies "___" should be a functor, but products aren't a functor, they're an object (or an object with some morphisms depending on which of my books i'm reading)
so should he instead be saying "the taking of products is representable"
It's really the universal property itself that constructs the functor $F$ I gave above (if you know about limits, you will see the role $A\to (A,A)$ plays in the product's definition).
i gotta start a problem by writing a definition and working up from there, but what he's written isn't definable from the definitions i've been given, in any of the books we're using
it's literally "Show that products are representible"
@KarlKronenfeld riiiight... i see, that is a good way to think of it
it's a shame i'm not more comfortable with some of the weird constructions in topology and whatnot, i think i'd have a better handle on some of this stuff if i knew more examples
i wonder if there's a big list of categories having to do with finite groups somewhere i could look at in the meantime
@AlexanderGruber I do not know, but I am reminded of Mac Lane referring to one of his works where he proves some of the basic facts of group theory from a category-theoretic stand point. Maybe that would be useful; now it's a matter of remembering where I read that. :)
@AlexanderGruber No, though I do love thinking about category theory. I am actually a self-studier with the goal of eventually working in algebraic geometry; however, I am in no rush.
@AlexanderGruber It's cool looking and I like both algebra and geometry. :) To sound less simplistic: it is very novel to me to even think of connecting quotients of polynomial rings with geometric figures and then generalizing.
if $lim_{x \rightarrow a} f(x) = L$, then prove that $lim_{x\rightarrow a}[f(x)-L] = 0, is it ok to do the proof $lim_{x\rightarrow}f(x) - \lim _{x\rightarrow} L = L - L = 0$?
$lim_{x\rightarrow}f(x) - \lim _{x\rightarrow} L = L - L = 0$
or is the proof asking for more than that?
$$\lim _{x\rightarrow a}$$
that's what I meant, sorry for the poor formatting
Let $x$ and $n$ be positive integers such that $1+x+x^2+\dots+x^{n-1}$ is a prime number. Then show that $n$ is a prime number. What is such prime called?
