All Questions
Tagged with lp-spaces metric-spaces
89
questions
2
votes
0
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32
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Linear Analysis – Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$
Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$?
I know how to prove for $p=2$ ...
- 451
0
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0
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44
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Trying to understand the proof for the criterion of compactness in $l_p$ space
I have the following theorem about the criterion of compactness in $l_p$ space
For the set $K\subset (l_p,||.||_p), p\geq 1$, following conditions
are equivalent:
i) $K$-totally bounded in $(l_p,||.||...
- 249
0
votes
1
answer
58
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Interpolation spaces
Hy everybody
I am curious about the definition; A compatible couple $(X_0, X_1)$ of Banach spaces consists of two Banach spaces $X_0$ and $X_1$ that are continuously embedded in the same Hausdorff ...
- 1,251
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.
...
- 3,482
2
votes
0
answers
65
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Use cases for $L^p$ and $l^p$ spaces where $p\neq 1,2,\infty$
Soft question: $L^1,L^2,$ and $L^\infty$ spaces all have many practical uses and an easy intuition behind them (Along with the $l^1$, etc. versions). Just for visualization, I was playing around ...
- 16.7k
2
votes
0
answers
49
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If $X$ is locally compact, then $\mathrm{Lib}_c (X)$ is dense in $L^p (X, \mu)$ for $p \in [1, \infty)$
Let $(X, d)$ be a metric space. Let $\mu$ be a Radon measure on $X$, i.e.,
$\mu$ is locally finite,
$\mu$ is tight on every Borel set, and
$\mu$ is outer regular on every Borel set.
As a result, $\...
- 17.6k
1
vote
0
answers
33
views
Is there a standard notational convention for $L_p$ ($\ell_p$) spaces, norms, and metrics?
In reading about $L_p$ spaces and $\ell_p$ norms and metrics, I have seen a wide variety of different notations for the same thing.
Is there a standard convention for when to use lowercase $\ell$ vs ...
- 2,454
0
votes
0
answers
128
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density of the set of sequences that eventually converges to 0
My teacher gave me a question and I don't know if my answer is correct (he didn't tell me :( )).
The question was : if I consider the set of all functions that eventually converges to 0, this set is ...
- 65
4
votes
0
answers
113
views
Relationship between the diameter of a set and the radius of a closed ball that contains it, in the $L_1$ space
What is the minimal radius $r(d)$ such that for every finite set of diameter $d$ there is a closed ball of radius $r(d)$ that contains it? This in the $L^1$ space with $n$ dimensions.
In formulas, ...
- 141
3
votes
0
answers
68
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A $\sigma$-algebra is complete w.r.t. the metric $d_{\mu}(A, B) := \mu(A \triangle B)$
I'm trying to prove below statement that appears in this question.
Let $(\Omega,\mathcal{A},\mu)$ be a finite measure space. We define a pseudometric $d_\mu:\mathcal A \times \mathcal A \to [0, \infty)...
- 17.6k
4
votes
0
answers
151
views
A finite measure space is separable if and only if its induced $L_1$ space is separable
I'm reading about symmetric difference operator on a $\sigma$-finite measure space from Wikipedia. Below is my try to prove a statement mentioned in this article.
Let $(\Omega,\mathcal{A},\mu)$ be a ...
- 17.6k
1
vote
1
answer
48
views
Extension of $L^p(\mathbb{R}^n)$ to a metric space
I know the following definition of $L^p(\mathbb{R}^n)$ with $1\leq p < \infty$: $$L^p(\mathbb{R}^n) := \left\{ f:\mathbb{R}^n \to \mathbb{C} : \left( \int_{\mathbb{R}^n}|f|^pd\mu \right)^{1/p} < ...
- 973
1
vote
1
answer
65
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How does completion affect the topology?
Given a set $X$ there are two ways to turn it into a topological space, first, specify the convergence (of nets), second, specify the topology. The two ways are equivalent.
Let $X=C(0,1)$ equipped ...
- 327
1
vote
0
answers
42
views
What shape has the largest area given perimeter ratio in $l_p $ space
Assume in 2-D $l_p$ space, The infinitesimal distance $ds=(|dx|^p+|dy|^p)^{1/p}$, and area $dA=dxdy$. What shape has the largest area/perimeter? My guess is the shape with largest area/perimeter in $...
- 87
1
vote
1
answer
66
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The space $\operatorname{Lip}_{b}(X)$ is dense in $L_1(X, \mu)$ w.r.t. $\|\cdot\|_{L_1}$
I'm trying to prove this well-known property. Could you verify if my attempt is fine?
Let $(X, d)$ be a metric space, $\operatorname{Lip}_{b}(X)$ the space of Lipschitz continuous bounded real-valued ...
- 5,817