All Questions
Tagged with lp-spaces banach-spaces
378
questions
3
votes
2
answers
92
views
Limit depending on parameter and $L^1$ function
What is the $\lim_{n\to\infty} n^a\int_0^1 \frac{f(x)dx}{1+n^2x^2}$ depending on $a\in\mathbb{R}$, if $f\in L^1(0,1)$?
By Banach-Steinhaus theorem I deduced that the limit is zero for $a\leq 0$, but I ...
- 6,515
2
votes
1
answer
62
views
For $1\le p < +\infty$ $L^p$ is a Banach space: Real and abstract analysis, Hewitt - Stromberg
I have some doubts about the proof of this theorem. From time to time I will put my justification.
For $1\le p < +\infty$, $L^p$ is a Banach space
Let $(f_n)_n$ be a Cauchy sequence in $L^p$, i.e., ...
- 47
2
votes
0
answers
71
views
Can one construct an isometric embedding $\phi_p:\ell^p\to\ell^\infty$ without Hahn-Banach?
It is well-known (for example, see this question) that, as a consequence of the Hahn-Banach theorem, every separable Banach space can be isometrically embedded in $\ell^\infty$. In particular, for $1\...
- 4,817
2
votes
0
answers
39
views
Can angles be defined by norms that are not induced by inner products?
Can angles be (well-)defined in a normed vector space where the parallelogram law does not hold? In other words, if the norm is not induced by an inner product in a normed space (say $L^1$ space), can ...
- 6,399
1
vote
2
answers
52
views
Computing the spectrum of the operator $A(f)(t)=tf(t), A:L^1([0,1])\to L^1([0,1])$
Let $A$ be the bounded linear operator from the Banach space $L^1([0,1])$ to $L^1([0,1])$ defined as $A(f)(t) = tf(t)$. I've come to learn that then $A$'s spectrum should be $[0,1]$, but I'm a bit ...
- 1,221
2
votes
2
answers
82
views
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
1
vote
1
answer
38
views
Minimizing vector for a closed set in $L^3$
Let $\Omega$ be a measurable set in $R$. $X =L^{3}(\Omega)$ and $Z$ be a closed subspace of $X$. Let $x \in X$. Show that $\exists y \in Z$ such that $\lVert x-y\rVert=\inf_{z \in Z} ||x-z||=\delta$.
...
- 307
2
votes
1
answer
90
views
Calculating operator norm of operator $(Ax)(k) = \int_{0}^{1} x(t)\operatorname{arcctg}\left((-1)^kt\right) \,\mathrm{d}t$
I have no idea how to solve this:
$$A: CL_1[0,1] \rightarrow \ell_\infty$$
$$(Ax)(k) = \int_{0}^{1} x(t)\operatorname{arcctg}\left((-1)^kt\right) \,\mathrm{d}t,\quad\forall x \in CL_1[0,1],\quad\...
0
votes
0
answers
73
views
Is it possible for a sequence in $L^p$ to converge to an element of $L^q$ ($q>p$) w.r.t. the $q$-norm?
Suppose that $1 \leq p < q \leq \infty$ and that $f_n$ is a sequence in $L^p(\mathbb{R})$.
If $f_n$ is a Cauchy sequence with respect to the metric induced by the $L^p$ norm, then (I believe that) ...
- 6,280
0
votes
1
answer
53
views
Weak star convergence in $L^{\infty}$ and strong convergence in $L^2$ imply strong convergence in $L^p$
I can't see why the following would hold. Can someone please point out why?
If $f_n$ convergence weak-* in $L^{\infty}(\Omega)$, and converges
strongly in $L^2(\Omega)$, then it must also converge ...
- 585
2
votes
1
answer
115
views
$\|f \otimes g\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^p}$ for $1 \leq p \leq \infty$
I am looking to prove the inequality
$$\|f \otimes g\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^p}$$
for $1 \leq p \leq \infty$. Here $f$ takes values in a Banach space $X_1$ and $g$ takes values in a Banach ...
- 6,275
3
votes
0
answers
28
views
If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^p_{\text{loc}} (Y)$, then $(f_n)$ is Cauchy in $(L^0 (Z), \rho_Z)$
Let
$T>0$ and $p \in [1, \infty)$,
$X :=[0, T]$ and $Y:= \mathbb R^d$,
$\cal A$ the Lebesgue $\sigma$-algebra of $X$,
$\cal B$ the Lebesgue $\sigma$-algebra of $Y$,
$\mu, \nu$ complete finite ...
- 5,817
0
votes
0
answers
28
views
Convergence in marginal measure implies that in product measure
Let
$T>0$ and $p \in [1, \infty)$,
$X :=[0, T]$ and $Y:= \mathbb R^d$,
$\cal A$ the Lebesgue $\sigma$-algebra of $X$,
$\cal B$ the Lebesgue $\sigma$-algebra of $Y$,
$\mu, \nu$ complete finite ...
- 5,817
1
vote
0
answers
51
views
Let $\|f_n-f\|_{L^p_{\text{loc}}} \to 0$. There is a subsequence $(n_k)$ such that $f_{n_k} \xrightarrow{k \to \infty} f$ a.e.
Let $p \in [1, \infty)$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $f:Y \to \mathbb R$ such that
$$
\|f\|_{L^p_{\text{loc}}} := \sup_{y \in Y} \|1_{B(y, 1)} ...
- 17.6k
2
votes
0
answers
34
views
Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Then the map $x \mapsto \|f(x, \cdot)\|_{L^p_{\text{loc}}}$ is measurable
Below we use Bochner measurability and Bochner integral. Let $T>0$ and $p \in [1, \infty)$. Let $X :=[0, T]$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $...
- 17.6k