1
vote
1answer
53 views

Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous?

Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous? ( A Banach function space is order-continuous $\equiv$ Increasing sequences of order-bounded positive functions ...
2
votes
0answers
33 views

Bochner measurability

I have the following problem. Let $(\Sigma, \Omega, \mu)$ be a measure space and let $X$ be a Banach space. Take a function $f \colon \Omega \rightarrow \mathbf{B}(X)$, which takes values in space of ...
0
votes
1answer
54 views

Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
0
votes
1answer
48 views

Density of linear span of idempotents in $L^{\infty}$

How do I show that the linear span of idempotents is dense in $L^{\infty}(\Omega,\mu)$ where $(\Omega,\mu)$ is a measure space? I don't really have any idea how to do this. Does it involve ...
0
votes
0answers
28 views

About weak-measurability in L(X,Y)

I'm starting a project about vector- (Banach) valued funtions and measures, and I know some of the basic definitions (measurable,weakly measurable...). The functions of study are ...
1
vote
1answer
48 views

Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
2
votes
1answer
53 views

A problem of maximization in Banach spaces

Let $X$ be a Banach space and $K\subset X$ compact. Let $C(K)$ be the set of continuous function in $K$ and $\mu\in (C(K))^\star$ a non-negative measure. Assume that $f:K\to X^\star$ is a continuous ...
2
votes
2answers
118 views

Linear combinations of delta measures

Let us consider the space of Borel, regular, complex measures on the real line, endowed with the total variation norm. Inside this space, I would like to characterize the space of all the finite ...
2
votes
0answers
37 views

Is space of Dirac measures Banach?

Is the space of all Dirac measures on a set $\Omega$ Banach? With the total variation norm. I don't know what convergence means in this norm.. I mean how do I even think about it.
0
votes
1answer
80 views

Supremum of measurable function

Let $X$ be a Banach space and for each $t \in [a,b]$ let $Y_t$ be a Banach space. Let $F_t:X \to Y_t$ be a bounded map for each $t$. I know that for given $u \in X^*$ and for all $w \in X$, ...
3
votes
1answer
69 views

Extend a linear functional on “nice” functions to an integral

I have a positive linear functional $h$ defined on a set of Lesbesgue-measurable functions of "moderate growth" on $\mathbb{R}^2$–call this set $MG(\mathbb{R}^2)$. (A function $f$ is positive if ...
5
votes
0answers
152 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
1
vote
1answer
115 views

What is the relationship between convergence uniformly, pointwisely, weakly, in $L^{\infty}$ norm and in $L^{p} $ norm?

What is the relationship between convergence uniformly, pointwise, weakly, in $L^{\infty}$ norm and in $L^{p}$ norm? I am quite puzzled by so many convergences, can anybody tell me what is the ...
1
vote
1answer
109 views

$L_p$ spaces and convergence

The Riesz-Fischer Theorem implies that Lp-convergence implies pointwise a.e. convergence of a subsequence. There is an example that shows that the converse may not be true... Let E = [0, 1], $1 ...
2
votes
0answers
81 views

Integration for functions with values in a separable Banach space

Let $(X,\mathcal{M},\mu)$ be a measure space, $Y$ a separable Banach space, and $L_{Y}$ the space of all $(\mathcal{M},\mathcal{B}_{Y})$-measurable maps from $X$ to $Y$ (where $\mathcal{B}$ denotes ...
2
votes
0answers
87 views

Group action and Radon measure

Let $\mathscr M(\mathbb R)$ be the Banach space of complex-valued Radon measures on $\mathbb R$, and let $\pi$ be the action of $\mathbb R$ on $\mathscr M(\mathbb R)$. Let $\mathscr A$ denote a subset ...
1
vote
0answers
59 views

A question about convergence in $L^p$. [duplicate]

Let $E$ be measurable and $1 \le p \le \infty$. Suppose $\{f_n\}_{n \in \mathbb{N}}$ all measurable and $\{f_n\}_{n \in \mathbb{N}} \to f$ pointwise a.e. $E$. For $p$ as above, I want to show that: ...
3
votes
2answers
258 views

Dual of $\ell_\infty(X)$

Given a Banach space $X$. Consider the space $\ell_\infty(X)$ which is the $\ell_\infty$-sum of countably many copies of $X$. Is there any accessible respresentation of the dual space ...
2
votes
1answer
85 views

Proving $L^2$ convergence (application of dominated convergence?)

For any $f\in L^2(\mathbb{R}^d)$ prove \begin{align}\left\lVert \int_{\mathbb{R}^d} e^{i |x-y|^2}f(y) dy-\int_{\mathbb{R}^d}e^{i |x-y|^2} e^{-|y|^2/a}f(y) dy \right\rVert_{L^2} \rightarrow 0\ \ \ ...
0
votes
1answer
94 views

Relationship between different$L^p (\Omega, \mathcal{F}, \mu)$ spaces with $\Omega$ uncountable and $\mu$ being a Radon measure

I earlier asked this question but I have not had a general classification in the posted answers there. So here is a new question. I am looking now for some special cases as suggested in one of the ...
0
votes
3answers
160 views

References on relationships between different $L^p$ spaces

I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.
4
votes
1answer
191 views

$L_p$ norm not subadditive for $0<p<1$ when endowed on $C[0,1]$

According to Wikipedia, the $L_p$-norm is not subadditive when $p\in(0,1)$. How can I show that the map $n_p(f)=(\int_0^1|f(x)|^p~\mathrm{d}x)^{2p}$ is not subadditive for $f\in C[0,1]$ for ...
5
votes
2answers
366 views

How to prove that the $L^p$ spaces are infinite dimensional

It is well-known that (given a measure space $(S,\mathcal A,\mu)$ and $1\le p\le\infty$) the Banach space $L^p(S,\mathcal A,\mu)$ has infinite dimension. Is there an easy way to proof this statement ...
12
votes
2answers
569 views

Isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?

Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$. Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
0
votes
1answer
90 views

How to `bound' $L^\infty$ by the constant function $1$

Let $(S,\mathcal A, \mu)$ be a measure space and consider the Riesz space $L^\infty=L^\infty(S,\mathcal A, \mu)$ (under point-wise ordering). Let $1_X$ denote the indicator function on $S$ (which is ...
3
votes
2answers
184 views

Shifting a function is continuous

I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
9
votes
1answer
902 views

Space of Complex Measures is Banach (proof?)

How can we prove that the space of Complex Measures is Complete? with the norm of Total Variation. I have stuck on the last part of the proof where I have to prove that the limit function of a Cauchy ...
2
votes
1answer
58 views

$a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map

Proof for $a\in (0,1)$ that $a\to F(a)= \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map, if $f\in L^p(X)$ for all $p\geq1$ in some measure space $(X,\mu)$. I'd like to prove that for all ...
3
votes
0answers
108 views

Question about proof of completeness of $L^p$

In my notes we prove completeness of $L^p$ by showing that if $\sum\|f_k\|_p < \infty$ then $\sum f_k$ converges in $L^p$. (That's a lemma we prove a bit earlier, namely that $(V, \|\cdot\|)$ is ...
6
votes
1answer
466 views

Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some ...
2
votes
1answer
223 views

How to show that these spaces are Banach spaces

I want to show, that the following spaces are Banach spaces: $X_1:=\{M=(M_t)_{0\le t \le T} ;\mbox{ M is an adapted RCLL process }\}$ with the norm $\|M\|_{X_1}:=\|\sup_{0\le t\le T}|M_t|\|_{L^2(P)}$ ...
1
vote
1answer
301 views

Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different ...
5
votes
2answers
135 views

$L^p$ space question

Assume $(X,\mathcal{M},\mu)$ is a measure space and for some $1\leq p<\infty$, $1\leq q<\infty$, $L^p(\mu)\subset L^q(\mu)$. Prove there is a constant $C>0$ so that $\|f\|_q\leq C\|f\|_p$ ...
2
votes
2answers
1k views

Prove that the normed space $L^{\infty}$ equipped with $\lVert\cdot\rVert_{\infty}$ is complete. [duplicate]

Possible Duplicate: Understanding proof of completeness of $L^{\infty}$ Most of the materials I have in Real Analysis consider this statement as a trivial one: "The normed space ...
1
vote
1answer
147 views

Compact, bounded sets and measures of non-compactness

Let $\gamma$ denote the Hausdorff/Kuratowski measure of noncompactness defined on a Banach space $(X,\|\cdot\|)$. I was wondering whether $\gamma(A)=\gamma(A+K)$ holds for $A\subset X$ is bounded and ...
4
votes
1answer
443 views

space of bounded measurable functions

Let $(\Omega, \Sigma)$ be a measurable space. Is the space of bounded measurable functions $B_b(\Sigma)$ equipped with the supremum norm a Banach space, i.e. complete?
2
votes
1answer
80 views

A question regarding convergence of distances to closed balls in Banach spaces

Let $X$ be Banach and let $B(x,\varepsilon)$ be the closed ball of radius $\varepsilon>0$ around $x\in X$ and consider the sequence $$f_{n;x}(y)= \begin{cases} 1-n\cdot d(yB(x,\varepsilon)), ...
4
votes
1answer
179 views

Measuring closed balls

Let $(X,\parallel \cdot \parallel)$ be Banach and $$\mathcal{BC}(X)=\{A\subset X\colon A \text{ is closed, bounded and non-empty}\}.$$ The natural metric on this space is the Hausdorff distance $d_H$ ...
6
votes
2answers
247 views

Definitions of measurability for operator-valued functions

If $\mathfrak{X}$ is a Banach space, a function $T: \mathbb{R} \to \mathcal{L}(\mathfrak{X})$ is defined to be uniformly measurable if it is an a.e. norm limit of a sequence of countably valued ...
7
votes
2answers
292 views

Bochner integral = 0 iff $f = 0$

This problem is about integrals of functions taking values in a Banach space. Let $f \in L^1(X,S,\mu,B)$ where $X$ is a set with a $\sigma$-algebra $S$ and a measure $\mu$. Function $f$ takes ...
4
votes
1answer
746 views

Is $p$-norm decreasing in $p$?

I could show that $\|\cdot\|_p$ is decreasing in $p$ for $p\in (0,\infty)$ in $\mathbb{R}^n$. Following are the details. Let $0<p<q$. We need to show that $\|x\|_p\ge \|x\|_q$, where $x\in ...
5
votes
2answers
691 views

Dual space of the space of finite measures

Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well ...
5
votes
2answers
347 views

Positive functionals on $\ell^\infty$

A continuous linear functional $\varphi: \ell^\infty \to \mathbb{R}$ is said to be positive if $$ x \ge 0 \rightarrow \varphi(x) \ge 0 \quad \forall x \in \ell^\infty.$$ If $\varphi$ is in $\ell^1 ...
49
votes
3answers
2k views

Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
5
votes
2answers
343 views

Weak limit of an $L^1$ sequence

We have functions $f_n\in L^1$ such that $\int f_ng$ has a limit for every $g\in L^\infty$. Does there exist a function $f\in L^1$ such that the limit equals $\int fg$? I think this is not true in ...