0
votes
0answers
20 views

If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
0
votes
2answers
76 views

Let $D$ be a subset of $L^2[0,1]$ defined in the following way

any help with this problem, it gave me hard time: Let $D$ be a subset of $L^2[0,1]$ defined in the following way: A function $f$ belongs to $D$ if and only if f is equal almost everywhere to a ...
4
votes
1answer
56 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
-1
votes
1answer
48 views

equi integrablity

See page 5 here. Let $\Omega$ be an open subset of $\mathbb{R}^n$, and let $(f_n)$ be a sequence of measurable functions, $f_n \in L^1(\Omega)$, which is bounded in $L^1(\Omega)$ ($f_n \in ...
0
votes
1answer
60 views

there is a measurable function $f$ on $X$ such that $|{f(x)}|=1$ for a.a $x \in X$ and $\nu(E)=\int_Efd|{\nu}|$ for any $E \in \mathfrak{M}$

any hints on this problem: Let $\nu$ be a finite signed measure on a measure space $(X, \mathfrak{M})$ and let $|{\nu}|$ be its total variation, prove that there is a measurable function $f$ on $X$ ...
0
votes
3answers
107 views

Hi I was thinking about a problem and have a question: [on hold]

Hi I was thinking about a problem and have a question: we know that if $f∈C([0,1])$ for which $∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$! Now my question is: Do we still have the same when we ...
3
votes
0answers
36 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
0
votes
1answer
39 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
0
votes
3answers
79 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
2
votes
1answer
40 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
0
votes
1answer
22 views

If $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$

Suppose $H$ is a Hilbert space. Is it true that if $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$ for any fixed $h\in H$? Certainly if $x_n\to ...
0
votes
1answer
26 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
1
vote
1answer
27 views

Extend projection on $L^2$ to one on $L^p$

if we have a closed subspace of $L^p$ called $X \cong l^2$ where the topologies of $L^p$ and $L^2$ coincide (we assume $p>2$). Then we can regard $X$ as a subspace of $L^2$, which means that he is ...
2
votes
1answer
39 views

Scalar products and partitions of Hypercubes

My questions relate to scalar products defined in $\mathbb{R}^{n}$ and partitions of hypercubes. Take $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$. My first question is why is it possible to ...
0
votes
0answers
56 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
1
vote
1answer
42 views

Weak*-convergence of probability measures

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable, bounded map. Let $(\mathbb Q_n)_{n\in\mathbb N}$ be a sequence of probability measures ...
5
votes
2answers
187 views

Borel Sigma Algebra

The question is asking to prove that the family: $\{(−a, a) : a \in \mathbb{R}\}$ does not generate the Borel $\sigma$-algebra. It is known that the family $\{(a,b) : a < b\}$ generates the Borel ...
1
vote
1answer
45 views

subspace of Hilbert space is closed if and only if it is weakly closed

Any hints for this question, thank you! Prove that a subspace of Hilbert space is closed if and only if it is weakly closed.
1
vote
1answer
66 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
0
votes
0answers
36 views

Notation in Lp spaces

I have a question about notation. If we have the space $L_p([a,b])$ with $1\leq p<\infty$ and $f\in L_p$ is it true that $\int_a^b \! |f(x)|^p \, \mathrm{d}x < \infty, \forall x\in[a,b]$. I ...
1
vote
0answers
39 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
1
vote
0answers
32 views

When is a delta function a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...
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 ...
0
votes
0answers
24 views

dose the weak topology change when we change the norm

I have a quick question, any help is appreciated dose the weak topology change when we change the norm? for example id we construct a norm on a linear space.
1
vote
1answer
37 views

If $u \in L^2(\Omega)$, then $\text{sign}u \in L^2(\Omega)$?

If $\Omega$ is a bounded domain and $u$ is in $L^2$, why is $\text{sign}(u) \in L^2?$ I am only stuck with the measurabilituy part. the integral is obviously finite on a bounded domain.
3
votes
1answer
23 views

bounded measure and dense subset of continuous functions

Let $C_0(R^n)$ be the space of continuous functions from $R^n$ to $R$ which vanish at infinity. Let $D$ be a subset of $C_0(R^n)$, I'd like to prove that if D is not dense in $C_0(R^n)$, then there ...
2
votes
1answer
56 views

Application of Riesz representation theorem

Suppose the following situation. We have linear functional $l$ on the space $H(\mathbb{C}^n)$ of entire function and wish to find a representation for $l$ with integration against a complex Borel ...
1
vote
0answers
41 views

Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement ...
1
vote
1answer
56 views

weak convergence of a bounded linear operator [duplicate]

I need help with this problem Let $X$ be a reflexive Banach space and $T: X \to X$ a linear operator. Show that $T$ belongs to $\mathcal{L}(X,X)$ if and only if whenever $\{x_n \}$ converges weakly ...
1
vote
1answer
60 views

Application of Riesz representation theorem and norm of linear functional.

I think the solution to this question somehow involves Riesz Representation Theorem, but I don't see how to apply it. Suppose $\{X,\mathcal{M},\mu\}$ is a $\sigma-$ finite measure space, $1\leq ...
1
vote
2answers
96 views

normed linear space of polynomials restricted to $[a, b]$

I have trouble with this problem Let $X$ be the normed linear space of polynomials restricted to $[a, b]$ . For $P \in X$, define $\phi(P)$ to be the sum of the coefficients of $P$. Show that $\phi$ ...
1
vote
1answer
34 views

convergence of a series in the space of bounded linear operator

I need help in showing that: If $X$ is a Banach space and $T \in L(X,X)$ have $||T||<1$. Use the completeness of $L(X,X)$ to show that $\sum_{n=0}^{\infty}T^n$ converges in $L(X,X)$. where ...
1
vote
1answer
18 views

Is a set of jointly bounded functions over a compact domain compact under p-norm?

Let $X$ be a metric space and a measurable space. Let $K$ be a compact set of nonzero measure and $r> 0$. Is a set $\{ f: K\rightarrow \mathbb R| |f|\leq r$ almost everywhere$\}$ compact with ...
0
votes
0answers
26 views

Problem about Projections

can someone help me with the following problem: Let $X$ be a Banach space and $P \in L( X,X)$ be a projection. Show that $P$ is open. where $L(X,X)$ is the space of bounded linear operators from ...
1
vote
0answers
20 views

Two different ways to generate the topology of convergence in measure

Consider the measure space $(X, \mathcal{B}, \mu)$ where $\mu(X) < \infty$. Let $L(X)$ denote the space of measurable functions on $X \rightarrow \mathbb{C}$. Then one way to define the topology ...
1
vote
1answer
60 views

an operator is it closed or bounded?

I need help with the following problem: Let $C^1[0,1]$ be the subspace of $C[0,1]$ that consists of continuously differentiable functions on [0,1], and let $A$ be the operator defined by $Af(x) = ...
2
votes
0answers
35 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
6
votes
1answer
56 views

Proof of equivalence of Sobolev Space and Lipschitz functions

The attachment is a proof from Evans book "Measure Theory and Fine Properties of Functions" pg 132 Theorem 5. The statement of the theorem is: Let $f:U \rightarrow \mathbb{R}$. Then $f$ is locally ...
0
votes
1answer
23 views

weak-*-convergence of measures ==> convergence of the total mass?

Let $X = [0,1]$. Let $\mu_n$ be a sequence of regular signed Borel measures on $X$, which converges to a measure $\mu$ on $X$ in weak-star, i.e. for any $f\in C_0(X)$, we have $\int_X f \mu_n(dx) \to ...
2
votes
0answers
47 views

Weak convergence in $L^1$

Does anyone have a reference for the following statement or similar ones? Let $U$ be an open bounded set in $\mathbb R^n$ and let $f\in C^0(U\times S^1)$. Then the sequence $f_m (x):=f(x,mx_i)$ ...
3
votes
1answer
93 views

$n^n$ are the moments of a measure on the non-negative real line?

I would like to know if the numbers $1,1,2^2,3^3,\dots, n^n,\dots$ are the moments with respect some measure $\mu$ on $[0,+\infty)$, i.e., if there exists such a measure $\mu$ with $$n^n=\int_0^\infty ...
0
votes
3answers
94 views

Measure Theory and Functional analysis exercise book

I'm looking for a big collection of exercises of functional analysis and measure theory. I know a lot of theory books which present some excercises (Brezis, Rudin, Lang, Royden, and others) but I was ...
0
votes
1answer
39 views

There exists a measure such that the sum of derivatives is the integral

This is a homework question in functional analysis. If $n \geq 1$, show that there is a measure $\mu$ on $[0,1]$ such that $\displaystyle\sum_{k=1}^n p^{(k)} \left( \dfrac{k}{n} \right) = \int p ...
0
votes
2answers
61 views

Borel measurable function

I'm struggling on the following question from a past paper: Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function and let $h:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined ...
0
votes
0answers
26 views

Fourier transform of a function of characteristic function of a measure

Let $\mu$ be complex measure on $\mathbb{R}^2$ ($|\mu|$ is finite measure) and $\chi$ - its characteristic function $$ \chi(x_1,x_2) = \int_{\mathbb{R}^2} d\mu(p_1,p_2) \exp(i p_1 x_1+i p_2 x_2). $$ ...
0
votes
1answer
48 views

Need a proof for an assumption on conditional probability density function based on probability theory

While reading book "Elements of Information Theory", I came across an assumption used in a proof on page 33. The assumption is as follows. Let $(X,Y)\sim p(x,y)=p(x)p(y|x)$. "If $p(y|x)$ is fixed, ...
1
vote
0answers
14 views

Distributions with a given mean and covariance

Fix $X := \mathbb R^d$ for some $d \ge 1$. Fix a vector $m \in X$ and a covariance operator $k : X^* \to X$, i.e., a symmetric, nonnegative-definite operator. Let $\Delta_{m,k}(X)$ denote the ...
3
votes
2answers
67 views

Lebesgue integral question from wiki

I have started studying Lebesgue integration and I have a question regarding the Lebesgue integral. In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: Let $f: ...
1
vote
1answer
40 views

A Borel measure defines semi-continuous function?

Let $X$ be a metric space with outer measure $\mu$, which is assumed to be a Borel measure, i.e., all Borel sets are measurable. For a fixed subset $A\subset X$ (not necessarily measurable, but you ...
1
vote
1answer
43 views

spectral measure of non-empty and open set is non-zero proof

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf Part d) on page 322 and his proof appears on page 324. I didn't quite understand his proof so I had a go at ...