All Questions
Tagged with lp-spaces functional-analysis
2,654
questions
4
votes
1
answer
81
views
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 ...
- 1,575
1
vote
3
answers
98
views
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 $$T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}...
- 520
1
vote
2
answers
119
views
Prove that $T$ is not a compact operator.
Let $T:\ell_{2}(\mathbb{Z})\rightarrow \ell_{2}(\mathbb{Z})$ be the operator defined by,
$$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 \...
- 520
0
votes
1
answer
53
views
$ L^p ( X ) \cap L^{\infty}( X) $ is a Banach space with respect only to the $p$-norm $\| \cdot \|_p$, $p<\infty$?
The space $๐ฟ^๐(๐) \cap ๐ฟ^\infty(๐)$, $p<\infty$, with the norm $||๐||_{๐ฟ^๐ \cap ๐ฟ^\infty}=||๐||_๐+||๐||_\infty$ is a Banach space. I imagine that if we remove the norm $||๐||_\infty$ ...
- 3,004
1
vote
2
answers
235
views
Is there a smooth function, which is in $L^1$, but not in$L^2$? [closed]
I am studying measure theory. While going over $L^p$-spaces I asked myself, whether there is $f\in C^\infty(\mathbb{R})$ s.t. $f\in L^1(\mathbb{R})\setminus L^2(\mathbb{R})$? I assume there could be ...
- 29
0
votes
0
answers
36
views
Spectrum of the laplacian outside of a compact
Let us consider $A$ a translation invariant lower semi-bounded operator on $L^2(\mathbb{R}^n)$ with domain $D(A)$ and with empty discrete spectrum (I exclude bound states). I have the following ...
- 57
2
votes
0
answers
32
views
Linear Analysis โ 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
votes
0
answers
38
views
Weighted $L^2$ space on Torus.
I'm studying weighted $L^2$ spaces in the circle $[0,2\pi]$
Definition 1
A weight is a function $w\colon [0,2\pi]\to \mathbb{R}^+$ (non negative)
Definition 2 The weighted $L_w^2([0,2\pi])$ is defined ...
- 3,650
0
votes
0
answers
26
views
Is there a Hilbert space of HenstockโKurzweil square-integrable integrable functions?
As is well-known, the space of square-integrable functions (say, on $[0,\,1]$) where the integral is a Riemann integral is not complete. If one completes it, one obtains the $L^{2}([0,\,1])$ Hilbert ...
- 221
0
votes
0
answers
27
views
Auxiliar inequality for Rellich-Kondrachov theorem
To prove the Rellich-Kondrachov Theorem it is used the following statement
If $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then $||\...
- 133
1
vote
2
answers
106
views
$L^{\infty}$ (uniform) decay of Dirichlet heat equation $u_t=\Delta u$
Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^N$. Consider the following initial-boundary value problem for the heat equation:
\begin{equation}
\begin{cases}
u_t=\Delta u\quad\quad\quad\;...
- 795
1
vote
0
answers
17
views
Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
- 2,612
0
votes
1
answer
34
views
$L_p$ inequality for measurable sets
Let $(U,\mu)$ a finite and positive measure space, and $1\leq p<\infty$. Suppose that for every $\varepsilon$ and measurable subset $A\subset U$, there exists a measurable subset $B\subset U$ such ...
0
votes
0
answers
27
views
Measurability of $\int_\Omega \varphi(x)u(t,x) \mathrm{d}x$ for $\varphi \in L^1(\Omega)$ and $u$ in a Bochner space
I have a function $u \in L^\infty((0,\infty), L^\infty(\Omega))$ where $\Omega$ is a bounded domain. Take $\varphi \in L^1(\Omega)$ and consider
$$f(t) := \int_\Omega \varphi(x)u(t,x) \mathrm{d}x.$$
...
- 670
2
votes
1
answer
54
views
How to interpret $L^2$ norm for functions from $[0,T]\to\mathbb{R}^n$?
I have a function $\alpha \in L^2(0,T;A)$ where $A\subseteq \mathbb{R}^n$.
I understand what it means when $A= \mathbb{R}$, i.e.
$$\Bigg(\int_0^T|\alpha(t)|^2dt\Bigg)^{1/2}<\infty$$
If $A\subseteq \...
- 23