# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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). ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 $w \in X$, ...
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### 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 ...
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### 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 ...
### 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 ...