Questions tagged [lp-spaces]
For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
5,707
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Limits of functions in $L^p$ spaces and Hölder inequality
I have a severe problem understanding $L^p$ spaces and everything related. For example, see my thoughts on the following exercise:
Let $f_n \in L^1(0,1) \cap L^2(0,1)$ for $n = 1, 2, 3, \ldots$ and ...
- 10.6k
0
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15
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estimation of different $L^p$ norms [closed]
I am wondering if it is possible to find a constant $C=C(p,T)$ such that
$\mathbb E[\int_0^T|Y_t|^p\mathrm{d} t]\le C(p,T) \mathbb E[\sup_{t\in [0,T]}|Y_t|^2],$
where $p>1$, $T$ some finite time ...
4
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1
answer
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weak convergence and pointwise implies $L_p$ convergence
Suppose $f_i \to f$ weakly in $L^p(X, M, \mu)$, $1 < p < \infty$, and that $f_i \to f$ pointwise $\mu$-a.e. Prove that $f_i^+ \to f^+$ and $f_i^- \to f^-$ weakly in $L^p$.
My proof:
Since $f^\pm ...
- 40.8k
1
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1
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Weak Convergence and Pointwise Convergence in L^p
Let $1<p<\infty.$
Let $U$ be a bounded open subset in $\mathbb R^n$.
Let ${u_k}$ and $u$ be functions in $L^p(U).$
Suppose $u_k\to u$ weakly in $L^p(U).$
There is a statement saying that the ...
- 40.8k
1
vote
2
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428
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Bounded linear functionals on $L^p(\mathbb{R})$, $0<p<1$.
Previously asked on this site: for $p\in(0,1)$, there are no bounded linear functionals on $L^p(\mathbb{R})$. I want to follow-up about what happens if we consider a general measure $\mu$; I do not ...
- 211k
0
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1
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How do we know the dual pairing between Lp spaces is well defined? [closed]
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
- 5,849
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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
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3
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Show that the linear functional is unbounded in $C_{00}$. defined as $T$ is defined as $T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}},$
Given a linear functional $T: C_{00}\to C_{00}$. Where $C_{00}$ is space sequences with finitely many non-zero terms with $\ell_2$ norm.
$T$ is defined as $$T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}...
- 33.9k
3
votes
2
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Strong smoothness of Lp norm
Define $f(x)=\|x\|_a^2$ for some $1<a<2$. It is well-known that (for example, Theorem 16 of http://ttic.uchicago.edu/~shai/papers/KakadeShalevTewari09.pdf)
$f$ is strongly convex w.r.t. $\|\cdot\...
- 2,292
1
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2
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Prove that $T$ is not a compact operator.
Let $T:\ell_{2}(\mathbb{Z})\rightarrow \ell_{2}(\mathbb{Z})$ be the operator defined by,
$$T((x_i)_{i\in \mathbb{Z}})=((y_i)_{i\in \mathbb{Z}}).$$
where
$$
y_{j}=\frac{x_{j}+x_{-j}}{2}, \quad j \in \...
- 40.8k
1
vote
1
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Show that $g_n(x)g_m(y)$ forms an orthonormal basis for $L^2(\Omega \times \Omega)$ when $g_n$ is an orthonormal basis of $L^2(\Omega)$
Let $\Omega \subset \mathbb{R}^n$. Show that $g_n(x)g_m(y)$ forms an orthonormal basis for $L^2(\Omega \times \Omega)$ for all $n,m \geq 1$ when $g_n:\Omega \rightarrow \mathbb{C}$ is an orthonormal ...
6
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2
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Integration is a compact operator on $L^p([0, 1])$
Let $p \in [1, \infty]$. I want to prove that this integral operator is compact:
$$
T_p: L^p([0, 1]) \to L^p([0, 1]), \quad T(f(x)) := \int_0^x f(t)dt
$$
I can prove it for $L_1$ case and I can prove ...
- 40.8k
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3
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$f_n \to f$ a.e. and $\| f_n\|_p \to \|f\|_p$. Is $\{f_n\}$ dominated by some $g$? [duplicate]
Let $E\subset \mathcal{M}(\mathbb{R}^n)$ with $m(E)>0$, $\{f_j\}_{j\in \mathbb{N}}\subset \mathcal{L}^p(E)$ and $f\in \mathcal{L}^p(E).$
Let $1\leq p < + \infty$ and suppose that $f_j\to f$ ...
0
votes
1
answer
53
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$ L^p ( X ) \cap L^{\infty}( X) $ is a Banach space with respect only to the $p$-norm $\| \cdot \|_p$, $p<\infty$?
The space $𝐿^����(𝑋) \cap 𝐿^\infty(𝑋)$, $p<\infty$, with the norm $||𝑓||_{𝐿^𝑝 \cap 𝐿^\infty}=||𝑓||_𝑝+||𝑓||_\infty$ is a Banach space. I imagine that if we remove the norm $||𝑓||_\infty$ ...
- 39.2k
3
votes
1
answer
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Eliminating Neumann boundary condition for elliptic PDE
In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:
I am now wondering if this also works with Neumann boundary ...
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