Tagged Questions
2
votes
1answer
61 views
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$)? And is one a subset of the other?
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$) ? And is one a subset of the other?
$\mu$ is the Lebesgue measure.
0
votes
0answers
26 views
Inclusion regarding the support of the convolution of two functions
Let $u \in L^1(\mathbb R^n)$ and $v \in L^p(\mathbb R^n)$ where $1 \le p \le \infty$. Show that $$\textrm{spt}(u \ast v) \subseteq \overline{\mathrm{spt}(u) + \mathrm{spt}(v)}$$ where the addition ...
1
vote
0answers
24 views
Estimation of a scalar product
I encountered the following, which shouldn't be that hard, but I can't get my head around it.
The problem is the following estimate (part of a bigger equation, but here's just the difficult part):
...
1
vote
1answer
95 views
Examples of $f \in L^p$ iff $p_0 < p < p_1$, $p_0 \le p \le p_1$ or $p = p_0$
Hi how to show the following:
Suppose $0 < p_0 < p_1 \leq \infty$. Find examples of functions $f$ on $(0,\infty)$
with Lebesgue measure such that $f \in L^p$
if and only if
(a) $p_0 < p ...
1
vote
0answers
61 views
Inclusion maps on $L^p$
How to show for $1 \leq p < q < r$
the inclusion maps
$$L^p \cap L^r \rightarrow L^q$$
$$L^q \rightarrow L^p + L^r$$
are continuous.
where the norms are defined in the following:
$L^p$ ...
0
votes
1answer
70 views
When is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$? [duplicate]
Given a measure space $(X,\Sigma,\mu)$, when is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$ for $p>r$, or for $p<r$ . Thanks.
0
votes
2answers
77 views
How to compute the norm of this particular bounded linear functional?
On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
1
vote
0answers
155 views
Rademacher function and weak convergence
The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
1
vote
1answer
97 views
Jensen's inequality and a estimate in $L^p$
In problem 3 we have:
If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and
$$\int_E e^{|f(x)|}dx =1,$$
then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and
...
3
votes
1answer
65 views
Index of a Fredholm Operator on Paths
I'm a novice to analysis but I need to understand the following example. Any help would be greatly appreciated. This might be of interest to some because it gives a way of quantifying changes in ...
0
votes
1answer
55 views
If f is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function g does that imply that f=0 a.e
If $f$ is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$
function $g$ does that imply that $f=0$ a.e
3
votes
2answers
138 views
Lp Spaces and limits of translated functions
If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$
where $g_{(t)}(x):=g(t+x)$.
Any hints? Try to give me only ...
12
votes
1answer
509 views
If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic
If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic
Maybe I would have to use the Rademachers.
