Hiroto Takahashi

?

Guys, there are any subspaces for $\ell^{2}$ such that $H_{s} \subset H_{t}$ such that $s < t$ for $0 \leq s, t \leq 1$

that sounds to Tychonoff's theorem

@TedShifrin Thanks!

just curious, there exist any infinite-dimensional compact space? please, don't judge me

$u_{n} \in H$*

I meant, if we have a sequence ${u_{b}} \in H$ where $H$ is a Hilbert space which converges weakly (i.e. $u_{n} \to u$) and $|u_{n}| \to |u|$, then $u_{n} \to u$ converges strongly? The answer here is false for a Banach space, but what about a Hilbert space?

The answer to this question, holds for Hilbert spaes?

I got a question

Hi there

@TedShifrin alright, it's fine.. that thing that I don't understand is $- \infty$

Hi someone can tell what does $g \in C^{- \infty}(\mathbb{S^{n-1}})$ mean? I was thinking that $g$ is a fast decreasing function on a sphere, but not sure..

is that mean that $g'$ is a fast decreasing function?

?*

why $-\infty$¿

let $g'$ be some distribution on the sphere $C^{- \infty}(\mathbb{S}^{n-1})$

somene can help me to understand this notation..

Hi there

@Huy it's Heine-Borel Theorem

@Huy I meant by definition

because if $X$ is compact then $X = \bigcup_{i=1}^{m} U_{i}$, then is clearly bounded and if we take the complement of X, that is, $\varline{X}$ we got $\emptyset$ which is strictily open in a bounded space, then $X$ is closed. Indeed, $X$ si compact. Right?

It follows inmediately that if a subest $U$ is bounded and closed then $U$ is compact, so I don't see any sense to that question.. or am I wrong?

I cannot see any example of that

It's possible to find a closed bounded equi-continuous subset of $C_{b}(\mathbb{R})$ that is not compact?

I have a Banach Spance $C_{b}(\mathbb{R})$ of bounded continuous real-valued functions on $\mathbb{R}$ with the supremum norm.

Hi guys

@DanielFischer Oh, thanks!

I know that a holomorphic function is a complex function which is differentiable

Hi, someone can explain what a holomorphic square root is intuitively?

@PedroTamaroff Thanks!

What does mean that: The metric on $\mathbb{C}$ restricts to the Euclidean metric on $\mathbb{R}$?

Hi, I have (maybe an easy) question

@SamuelYusim Thanks!

someone know an example about a groupoid that is not a group?

Hi everyone

$\{{1, (23)}\}$ is a non normal subgroup from $S_{3}$

Good afternoon from here!

Hi, someone can give an example about this: Let $\varphi: G \longrightarrow G'$ be a homomorphis group. If $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is a normal subgroup of $G'$?

@TobiasKildetoft yes, you right. sorry.

it's beucase I was thinking period of $2\pi$

@TobiasKildetoft Well, the sine curve is symmetric with respect to the $x$-axis

@TobiasKildetoft maybe you right

@TobiasKildetoft Well, It's symmetric with respect to $y$-axis

@TobiasKildetoft a parabola with vertex (0, 0) then

@TobiasKildetoft No :( can you explain me, please?

@MikeMiller I think is related to periods of sine. Like a cyclic group?

@TobiasKildetoft it's because I was thinking in the above group.

@TobiasKildetoft I have this problem: . Describe the group of isometries of the sine curve (the graph of $y = \sin x $): list its elements and construct a (compact) multiplication table.