All Questions
Tagged with lp-spaces convergence-divergence
372
questions
-1
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0
answers
26
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Why are limits of $L^p$ sequences defined almost surely? [closed]
I have heard it said that if a sequence of random variables $\{X_n\}$ converges in $ L^p $, then it converges to a limit $ X $ that is defined almost surely. I am trying to understand the precise ...
- 135
2
votes
0
answers
27
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If $h_k\to h$ in $L^2(\mathbb{R}^n$ then $(h_k\ast \varphi_{\varepsilon_k})\to h$ in$L^2(\mathbb{R}^n)$
Let $(h_k)_{k\in \mathbb{R}^n}\subset L^2(\mathbb{R}^n)$ converge to some $h$ in $L^2(\mathbb{R}^n)$ and $(\varphi_{\varepsilon_ k})_{k\in \mathbb{N}}$ be the standard mollification sequence. I want ...
- 890
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votes
1
answer
30
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Convergence in $L^\infty$($\Omega$) and almost everywhere
I have a question about the difference between the convergence in $L^\infty$ and convergence almost everywhere.
Precisly, let $\mu(\Omega) < \infty$, $f_n \rightarrow f$ in $L^1(\Omega)$, then ...
- 11
1
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0
answers
32
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Convergence on $L^1$ space
I come across this problem while reading hints for Exercise 3.15 (Brezis):
Let $\Omega=(0,1)$ and sequence $f_n$ defined by $f_n(x)=ne^{-nx}$. Prove that:
$$\displaystyle \int_{\Omega} \varphi f_n\...
0
votes
1
answer
60
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Convergence in $L^p_{loc}$ implies convergence of a subsequence in $L^\infty$
Let $\Omega \subset \mathbb{R}^n$ be bounded or unbounded. Suppose we have a sequence $\{f_n\} \in L^p_{loc}(\Omega)$ such that $f_n \rightarrow f$ in $L^p_{loc}(\Omega)$ for $f \in L^p_{loc}(\Omega)$....
- 6,275
1
vote
2
answers
180
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Does $\sin (nx)$ converge in $L^2$?
I was just introduced the concept that if $(f_n)$ converges in $L^2$ topology to $g(x)\in L^2([0,2\pi])$ then $\lim_{n\to\infty}\int^{2\pi}_0|f_n(x)-g(x)|^2dx=0$. I would appreciate any hint to how to ...
- 191
2
votes
1
answer
44
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Interchanging limits and integrals for Cauchy sequences in $L^p$
NOTE: $U\subseteq \mathbb R^n$ is an open, simply connected set.
I am reading L.C Evans's PDE book. I am on page 263. In order to show that Sobolev spaces are a kind of Banach space, we need to show ...
- 12.5k
1
vote
1
answer
60
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Why almost sure convergence cannot imply convergence in $L^p$? [closed]
Almost sure convergence cannot imply convergence in $L^p$. However I can't find a rigorous counterexample?
Can anybody help me? Any helpful ideas would be greatly appreciated!
- 23
1
vote
1
answer
95
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Brezis' exercise 8.28.11: how to prove $\sum_{k=0}^{\infty} \left | \frac{1}{2} \alpha_k (f) - a \right |^2 < + \infty$?
Let $I$ be the open interval $(0, 1)$ and $H := L^2 (I)$ equipped with the usual inner product $\langle \cdot, \cdot \rangle$. Consider the linear map $T: H \to H$ defined by
$$
(Tf) (x) = \int_0^x t ...
- 17.6k
1
vote
0
answers
111
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A question about probability theory and a little functional analysis
Consider the sequences of mean zero random variables $(X_n)_{n \in \mathbb N}$ with finite variances, i.e., $E[X_n]=0$ and $E\left[X_n^2\right] =: \sigma_n< \infty$, for each $n$. Moreover, ...
- 75
0
votes
0
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73
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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
0
answers
28
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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
0
votes
0
answers
26
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If $\rho_{X} (g_n, 0_\mathbb R) \to 0$ then $\rho_Z (f_n, 0_E) \to 0$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of ...
- 5,817
0
votes
0
answers
59
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convergence of $\int_{-1}^1\sin(n\pi x)f(x)dx$ to $0$
I was playing around with this question in my graphic calculator and noticed, that $\int_{-1}^1\sin(n\pi x)f(x)dx\to0$ for $n\to\infty$, where $f\in L^p((-1,1))$ and $1<p<\infty$. While trying ...
- 133
2
votes
1
answer
64
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Showing that $f(\lambda x)\to f(x)$ in $L^4$
I would like to show that for $f\in L^4(\mathbb{R})$, we have that
$$\lim_{\lambda\to 1} \int |f(\lambda x)-f|^4\,dx=0. $$
I first used the fact that $C_0^\infty$ is dense in $L^4$ so that, letting $\...
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