All Questions
Tagged with metric-spaces banach-spaces
184
questions
2
votes
2
answers
82
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Reference request: $\ell^1$, $\ell^2$, $\ell^p$
I would like to recive from you some references (books and or good notes) about the spaces $\ell^p$. I have searched over here already, but I really didn't find any good match to what I am asking for.
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- 3,492
1
vote
0
answers
36
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Updated Gorelik principle
One version of the Gorelik principle is the following: Let $E,X$ be Banach spaces and suppose $U:E\to X$ is a Lipschitz isomorphism (that is, a Lipschitz bijection whose inverse is also Lipschitz). ...
user1260017
0
votes
0
answers
35
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Completetion of Banach spaces for uncountable index
Let $I$ be an index set, and $V_i$ is a collection of Banach spaces. Consider $p < \infty$, I know that $\sum V_i$ is may not complete under the norm $$||f|| := \left(\sum ||\pi_i(f)||^p\right)^{1/...
- 629
3
votes
1
answer
132
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Every metric space can be embedded isometrically into the Banach space
I've been trying to construct my own proof of this, but the only reference I've been able to find that I could (somewhat) understand was the following. In particular, I'm confused at the use of $f_{a}(...
- 91
1
vote
0
answers
61
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Is there an isometric embedding from the Euclidean plane to the sequence space with the $\ell_1$ metric?
I am considering two metric spaces:
The Euclidean plane $\mathbb R^2$, equipped with the Euclidean distance metric $d((x, y), (x^\prime, y^\prime)) = \sqrt{(x - x^\prime)^2 + (y - y^\prime)^2}$, and
...
- 115
1
vote
1
answer
87
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Prove that the Faber-Schauder System is a basis for $C[0,1]$ with the sup norm.
I am trying to prove that the Faber-Schauder System is a basis for $C[0,1]$ but I am stuck at some point. I will define the relevant functions to my question and explain what I have done so far.
Let $...
- 115
0
votes
0
answers
25
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For a bijection $T\in\mathcal B(B_1, B_2)$ prove $B_1\setminus \text{Ker}(T)\cong \text{range}(T)$ iff $\text{range}(T)$ is closed.
Problem: Let $B_1$ and $B_2$ be two Banach spaces and take $T\in\mathcal B(B_1, B_2)$ to a bounded linear operator between the two that is also bijective. Prove that $B_1 \setminus \text{Ker}(T)\cong \...
- 1,222
0
votes
1
answer
43
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Baire's Theorem Rudin's RCA
Now I'm reading Rudin's RCA.
There I've readen Baire's Theorem.
There it is:
If $X$ is a complete metric space, the intersection of every coutanble collection of dense open subsets of $X$ is dense in $...
- 1,020
0
votes
0
answers
26
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Can we defined a metric so that the non-constant cauchy sequence do not exists, then the space is complete vacuiously
I know that not all metrics are induced by norm, only those metrics satisfy the homogeneity and translation invariant.
1. Now, Can a normed space equipped with metric not be induced by the norm?
2. If ...
- 1,364
2
votes
2
answers
88
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Are equivalence classes of norms that make the space complete unique?
In my real analysis two class, we are talking about how $C([a,b],R)$ is the space of continuous functions from some real interval $[a,b]$ to the real numbers. We have talked about how a normed space ...
- 21
3
votes
0
answers
143
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Which Banach spaces admit medians?
In a metric space $X$, let $I(x,y)$ denote the metric interval between two points, i.e. $I(x,y):=\{z:d(z,x)+d(z,y)=d(x,y)\}$. Given a triplet $(x,y,z)$, we say that $w$ is a median of the triplet if $...
- 1,098
1
vote
0
answers
54
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A generalization of Theorem 1.40 in Rudin's Real and Complex Analysis
Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Then we have Theorem 1.40 in Rudin's Real and Complex Analysis, i.e.,
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- 17.6k
2
votes
0
answers
193
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Let $(X, d)$ be a separable metric space. Then the space of all real-valued bounded uniformly continuous functions on $X$ is separable
It's well-known that
Theorem 1 A metric space is compact IFF its space of bounded, continuous, real-valued functions is separable in the uniform topology.
I would like to prove its complementary ...
- 5,817
2
votes
0
answers
66
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If $X$ is locally compact separable, then $\mathcal C_c(X)$ is dense in $\big (\mathcal L_p, \|\cdot\|_{\mathcal L_p} \big)$ for all $p\in [1,\infty)$
Let $X$ be a metric space, $\mu$ a $\sigma$-finite Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ be its semi-...
- 5,817
1
vote
1
answer
51
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The normed space of functions such that $x \mapsto \frac{f(x)}{x}$ is integrable
I was looking over some functional analysis problem sheets from Oxford and question 6 on this one caught my attention. I will copy it below:
Let $I=[0,2]$ and consider $$X=\{f\in L^1(I):\int_I\frac{1}...
- 703