0
votes
1answer
46 views

Borel Measure on Banach Space

While thinking about what some measure on an infinite dimensional Banach space could look like a came across the point that if I'd like to assign a size to all epsilon balls, they by Riesz' lemma ...
4
votes
1answer
79 views

Proof that $(L^1)\neq(L^\infty)^\ast$

I have seen a "proof" that $L^1\neq(L^\infty)^\ast$ which goes as follows: show that there is an element of $(L^\infty)^\ast$ which is not in the image of the canonical map ...
14
votes
0answers
323 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
0
votes
0answers
36 views

The restriction of an open bounded linear operator

I need some help with this question. Let $X$ be a Banach space and $T:X \to X$ be a bounded linear operator. Suppose that $T$ is open, and $X_0$ be a closed subspace of $X$. The restriction $T_0$ of ...
1
vote
1answer
32 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
2
votes
1answer
55 views

the point where all functional are non zero

Let $\{f_n\}$ be sequence of non zero bounded linear functionals on a Banach space X. Show that there is $x\in X$ so that $f_n(x)\ne0$, for all $n\in \Bbb N$. I am confused, non zero functional ...
1
vote
0answers
25 views

A question about uniform convergence in proof that $L^\infty$ is Banach.

I was reading this post Understanding proof of completeness of $L^{\infty}$ and it is mentioned that the sequence $(f_n)_{n\in\mathbb N}$ converges uniformly on a conegligible set $N^C$. Could someone ...
2
votes
0answers
42 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
2
votes
1answer
51 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
1
vote
1answer
51 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
2
votes
1answer
39 views

Banach space under the Lip norm

Let $(X,d)$ be a compact metric space. A function $f:x\to \Bbb R$ is said to be Lipschitz continuous if $$\|f\|_d = \sup\left\{\frac {|f(x)-f(y)|}{d(x,y)}:x,y\in X,x\neq y\right\}< \infty.$$ Denote ...
1
vote
1answer
80 views

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

Which are the conditions for a Lorentz space $L^{p,q}$ to be ord. continuous? ( A Banach function space is o-c $\equiv$ Increasing sequences of order-bounded positive functions converge in norm). ...
2
votes
0answers
37 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
61 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
60 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
29 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
52 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
56 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
149 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
45 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
84 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
74 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
159 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
121 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
126 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 ...
3
votes
0answers
95 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
95 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
282 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
86 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
100 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
177 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
207 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
436 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
588 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
94 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
193 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
1k 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
112 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 ...
7
votes
1answer
488 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
226 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
352 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
136 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
2k 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
154 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
477 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
83 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
180 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
269 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 ...