Tagged Questions

2
votes
1answer
30 views

Intuition behind closed subsets of a metric space?

Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space. Consider a metric space $$(X,d)$$ Then consider a subset of this space$$F$$ What the book ...
1
vote
1answer
46 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
1
vote
2answers
90 views

Continuous functions uniformly convergent to a function, metric spaces, equivalent conditions

Let $X, \ (Y, d)$ be metric spaces, $f_1, f_2, \ldots \ : X \rightarrow Y$ be continuous functions, $f: X \rightarrow Y$ an arbitrary function. Prove that the following condtions are equivalent: 1) ...
0
votes
2answers
49 views

Find a convergent function in metric space

Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$. Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
2
votes
0answers
45 views

convergence in metric space

Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$. Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
2
votes
2answers
22 views

Equality of limits with respect to different metrics.

Suppose that $X$ is a set equipped with two metrics, say $d_1$ and $d_2$. Let $\{x_n\}_{n\in\mathbb{N}}\subset X$ be a sequence of points which converges to $x\in X$ with respect to metric $d_1$. ...
1
vote
2answers
38 views

If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.

A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$ Let $(X,d)$ be a metric space and let ...
1
vote
1answer
26 views

Is a 'normally' convergent sequence still convergent in a metric space which barely excludes its 'normal' limit?

For example, suppose $$ x_n = \frac 1n \\ X = (0, 1)$$ Is $x_n$ convergent in $X$? My guess would be no, since there exists no $x \in X$ which $x_n$ approaches; $x_n$ will eventually surpass any ...
2
votes
1answer
31 views

Correctness of Converging sequence and Adherent Points

$x\in X$ is an adherent point of $A\subset X$ if for every $\epsilon>0$ there exists $y\in A$ s.t. $y\in B(x, \epsilon)$ $B(x, \epsilon)$ is the open ball centered at $x$ with radius $\epsilon$ ...
2
votes
2answers
94 views

Kernel of $p$-adic logarithm.

I'm completely clueless as to how to answer the following question: Let $K$ be a field of characteristic zero which is complete with respect to a non-Archimedean aboslute value $|\cdot|$. Let ...
2
votes
1answer
64 views

Subspace $Y$ of metric space with finitely many points is complete.

Show that if a subspace $Y$ of a metric space consists of finitely many points, then $Y$ is complete. This is what I have so far, but I don't know where to go from here: Suppose the the subspace ...
1
vote
1answer
141 views

Show $\mathbb R^n$ is complete.

Show $\mathbb R^n$ is complete. At this point, I am trying to work through the problem in my textbook, there is one step that I do not understand and would like explained. Here's my proof so far: ...
2
votes
1answer
61 views

Showing convergence in Space of Squared Summable Sequences

My Problem: Show that the sequence ${x_n}_{n\geq 1}$, where $x_n=(1,\frac{1}{2},\ldots,\frac{1}{n},0,0,\ldots)$ converges to $x=(1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n},\ldots)$ in $l_2$ My ...
2
votes
2answers
65 views

When and how does this sequence converge?

I cannot prove this statement, I tried to prove by using the definition of open sets however i feel that it is necessary prove it in two directions since it's an iff statement. The question is, Let ...
2
votes
1answer
120 views

Why is $L^3$ weaker than $L^2$?

Someone told me today that if I can show $\Vert A_n-B_n\Vert_3\to 0$ as $n\to \infty$, then claiming $A=B$ as $n\to \infty$ (where $A$ and $B$ are the respective limits of $A_n$ and $B_n$) is a weaker ...
0
votes
0answers
65 views

Convergence in Skorokhod metric and unifrom metric

Is there a relationship between convergence in the Skorokhod space and convergence in the uniform metric. I.e. does weak convergence in the Skorokhod space imply convergence in the uniform metric?
9
votes
3answers
439 views

In what spaces does the Bolzano-Weierstrass theorem hold?

The Bolzano-Weierstrass theorem says that every bounded sequence in $\Bbb R^n$ contains a convergent subsequence. The proof in Wikipedia evidently doesn't go through for an infinite-dimensional space, ...
2
votes
0answers
99 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
1
vote
1answer
162 views

Cauchy nets in a metric space

Say that a net $a_i$ in a metric space is cauchy if for every $\epsilon > 0$ there exists $I$ such that for all $i, j \geq I$ one has $d(a_i,a_j) \leq \epsilon$. If the metric space is complete, ...
1
vote
1answer
90 views

Uniform convergence of functions, Spring 2002

The question I have in mind is (see here, page 60, the solution is at page 297): Assume $f_{n}$ is a sequence of functions from a metric space $X$ to $Y$. Suppose $f_{n}\rightarrow f$ uniformly and ...
4
votes
2answers
102 views

Is Completeness intrinsic to a space?

Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric? If ...
1
vote
2answers
64 views

Is the set $E$ of sequences containing only entries $0$ and $1$ in $(m,\left \| \cdot \right \|_\infty)$ complete?

I can't really wrap my head around $E$, or a Cauchy sequence in $E$. I need to take a Cauchy sequence in $E$ and show it's Cauchy in $(m,\left \| \cdot \right \|_\infty)$? I think I can show $(m,\left ...
1
vote
3answers
110 views

continuous map of metric spaces and compactness

Let $f:X\rightarrow Y$ be a continuous map of metric spaces. Show that if $A\subseteq X$ is compact, then $f(A)\subseteq Y$ is compact. I am using this theorem: If $A\subseteq X$ is sequentially ...
8
votes
3answers
277 views

Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
2
votes
1answer
135 views

A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
6
votes
4answers
379 views

Convergence in metric and a.e

How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?
3
votes
2answers
138 views

How to prove that the sequence of $x_n = (1,\frac{1}{2}, \frac{1}{3}, … \frac{1}{n}, 0, 0…)$ does not converge under $\|\cdot\|_1$?

I'm reviewing past assignments and am still having trouble formulating a proof for this: Consider the sequence $(x_n)$, where $x_n = (1,\frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, 0, 0, \ldots)$. ...
6
votes
3answers
1k views

Examples of function sequences in C[0,1] that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
3
votes
3answers
453 views

How to prove that convergence is equivalent to pointwise convergence in $C[0,1]$ with the integral norm?

I'm trying to prove (or disprove) that in the set $C[0,1]$ of continuous (bounded) functions on the real interval [0,1] with the integral norm $\|f(x)\|_1 = \int_0^1|f(x)|dx$ that a sequence of ...
3
votes
1answer
102 views

If no Cauchy subsequence exists, must a uniformly separated subsequence exist?

Given a sequence $(x_n)$ in a metric space $M$, call it uniformly separated if all pairwise distances $d(x_n,x_m)$ between distinct terms are uniformly bounded away from zero. Suppose that a given ...