3
votes
0answers
26 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
5
votes
1answer
68 views

Derivative of $\frac{x}{f(x)}\frac{df}{dx}$

Suppose we have a function $f(x):\mathbb R^+\to\mathbb R^+$ that satisfies: 1) $0\leq\frac{df}{dx} \leq 1$ 2) $f(0) = 0$, then do we have $$\frac{d}{dx}\left(\frac{x}{f(x)}\frac{df}{dx} ...
5
votes
1answer
42 views

When is a vector space of polynomials dense in $C([0,1])$?

Weierstrass' theorem states, in particular, that the set of polynomials with real coefficients is dense (with the supremum norm) in the set of continuous function on $[0,1]$. Using the ...
1
vote
1answer
39 views

Linear operator in $\ell^2$

Let $A \colon \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be the linear operator defined by $\left( Ax \right)_k = \sum_{i \in \mathbb{Z}}a_{ki}x_i$, where $a_{ki} = 1/(k-i)^2$ if $k \neq i$ and ...
3
votes
1answer
51 views

Small $\ell^p$ spaces are obtainable from $L^p$

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, ...
1
vote
1answer
19 views

Books with exercises or problems on the space of functions of bounded variation

I am studying BV space (the space of functions of bounded variation) by using Evans & Gariepy's book. However, there are no exercises for BV space, and I have no idea where I can find some. ...
1
vote
2answers
40 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
2
votes
2answers
41 views

If $f(x)g(y)$ is a measurable function, and $f$, $g \in L^{1}(dm)$, does this imply $g(y - x) \in L^{1}(dm)$?

Question rephrased Suppose we are working in $(\mathbb{R}, \Sigma(m^{*}) \times \Sigma(m^{*}), m \times m)$ where $m$ is Lebesgue measure. Note that our $\sigma$-algebra is not necessarily complete. ...
0
votes
1answer
50 views

Question about statement of Fubini's theorem

This is a question on the statement of Fubini's theorem for measurable sets. The theorem looks like this: Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda = \mu \times \nu)$ be a complete ...
0
votes
3answers
43 views

If $a_n\ge nb_n$ and the sequence $(b_n)$ is unbounded, then the differences $a_{n+1}-a_n$ are also unbounded

Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq1}$ be sequences of positive numbers such that $a_n\geq n b_n$ for all $n >1$. Prove that if $(a_n)_{n\geq 1}$ is increasing and $(b_n)_{n\geq 1}$ is ...
3
votes
1answer
36 views

If $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, does that imply $(X, \Sigma, \mu)$ is $\sigma$-finite?

I'm having trouble proving or disproving the statement: If the product space $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, then so is $(X, \Sigma, \mu)$. I ...
2
votes
1answer
64 views

What exactly is a product measure?

If we have $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ a complete measure space with underlying complete spaces $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$, and $\lambda = \mu \times \nu$, what ...
0
votes
1answer
32 views

Dense subset of $L^{2}$ such that $x^{-1/2}f \in L^{1}$ and $\int_{[0, 1]}x^{-1/2}f\, dx = 0$

Does there exist a dense set of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that $\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
1
vote
1answer
31 views

Does the Fourier coefficients of a function $f\in H^1(0,L)$ (the first order Sobolev space) are absolutely summable?

My precise question: Let $f\in H^1(0,L)$ and let $\{f_n\}$ be its Fourier sine series coefficients on $(0,L)$, is it true that $\{f_n\}\in l^1$, i.e. $$\sum_{n}|f_n|< \infty .$$ Thanks
2
votes
1answer
31 views

If $f \in L^{1}(d\mu)$, is it true that $\int \limits_{X} f\chi_{\{ f \neq 0 \} } \,d\mu = \int \limits_{X}f \,d(\chi_{\{ f\neq 0 \} }\,d\mu)$?

Ok, so we have $f \in L^{1}(d\mu)$, with $(X, \Sigma, \mu)$ a complete measure space. If we assume $f$ is nonnegative, we can define a measure $\rho(E) = \int \limits_{E} f \,d\mu$ for $E \in ...
3
votes
1answer
40 views

If $f \in L^{1}(d\mu)$ is nonnegative, can we conclude $\mu( \{ x \mid f(x) \neq 0 \} ) < \infty$?

I am trying to prove a statement, and I need the fact that: If $f \in L^{1}(d\mu)$ is a nonnegative function, then this implies $\mu( \{x \mid f(x) \neq 0 \} ) < \infty$. But I don't know ...
0
votes
2answers
42 views

Integrable function with given condition is in $L^p$

Suppose $f:\Bbb R \to \Bbb R$ is integrable and there exist constant $c\gt 0$ and $\alpha \in (0,1)$ such $$\int_A |f(x)|dx\le cm(A)^\alpha$$ for every Borel measurable set $A\subset \Bbb R,$ where ...
1
vote
2answers
46 views

Showing a sequence is in $\ell^2$ [duplicate]

I am working on the following problem. Suppose that $\{a_j\}_{j=1}^{\infty}$ is a sequence with the property that, whenever $\{b_j\}_{j=1}^{\infty} \in \ell^2$, one has ...
6
votes
1answer
125 views

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
1
vote
1answer
33 views

Norm of the multiplication operator $f\mapsto (x\mapsto xf(x))$ on $L^2[a,b]$ [duplicate]

We have a linear operator $T : L^2[a,b] \rightarrow L^2[a,b]$ (with $|a| \le |b|$), $f \mapsto (x \mapsto xf(x))$ Now I shall determine what $\Vert T\Vert$ is. We clearly have $\Vert x \mapsto ...
3
votes
0answers
27 views

Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent. I have trouble to prove this, please help.Thanks Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ ...
1
vote
1answer
28 views

Prove that a linear and continuous operator admits inverse in Hilbert space

Let $(H,(\cdot,\cdot))$ an Hilbert space and $A:H\rightarrow H$ a linear and continuous operator such that there exists $\alpha >0$ such that $$(Au,u)\geq \alpha \|u\|^2 \text{ for each } u\in H.$$ ...
0
votes
0answers
92 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
3
votes
1answer
81 views

Questions on Fubini's Theorem and $\sigma$-finite measure?

I asked a question about this a several days ago, but I think I have a better formulated question now. The reason I did not just edit the last question about this is that I feel the answers I got ...
1
vote
1answer
22 views

Unit ball of continuous functions is a closed set - Proof with neighborhood argument

This question is trivial if one uses sequence definition, but I want to use the usual topological definition of closed set. That is , a set is closed if its complement is open. Let $U=\{f\in ...
4
votes
1answer
48 views

Compact closure in $C([0,2])$

a) Does the closure of $\left\{f_n(x)=\sin(x^n):n=1,2,3\dots\right\}$ form the a compact subset of $C([0,2])?$ b) Does the closure of $\left\{f_n(x)=\sin(x^\frac1n):n=1,2,3\dots\right\}$ form the a ...
0
votes
1answer
46 views

Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
2
votes
1answer
45 views

Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} ...
4
votes
1answer
80 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 ...
0
votes
1answer
28 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
1
vote
2answers
57 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
1
vote
1answer
58 views

Is the space of continuous and bijective functions $f\colon [0,1] \to [0,1]$ complete?

Let $X$ be the space of continuous and bijective functions $f$, such that $$ f\colon [0,1] \to [0,1] \quad , \quad f(0)=0 \quad , \quad f(1)=1 \, .$$ Is $X$ complete (under the supremum norm $ ...
1
vote
0answers
46 views

Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$ \check{\hat{f}}=\hat{\check{f}}, $$ where $$ \hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x $$ and ...
0
votes
1answer
44 views

There does not exist $f\in (l^\infty)^*$ with $\ker f = c_0$

There does not exist $f\in (l^\infty)^*$ with $\ker f = c_0$. $c_0$ is the closed subspace of $l^\infty$ with the property that if $x = (x_1, x_2,...) \in c_0$ then $$\lim_{n\rightarrow \infty }x_n = ...
1
vote
1answer
59 views

How do you show $\int \limits_{X \times Y} f(x,y)\, d\lambda < \infty$ if $\int \limits_{X} \int \limits_{Y} f(x,y) \,d\nu \,d\mu < \infty$?

Suppose $f: X\times Y \rightarrow [0,\infty]$ is a measurable function with respect to the product measure $\lambda$ ( $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are complete measure spaces). Suppose ...
2
votes
1answer
38 views

Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$ ||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}. $$ Let $\{\Omega_n\}$ be a sequence of open ...
2
votes
0answers
42 views

Taking limit and $\sup$

I was reading a proof for $l^\infty(\mathbb{N})$ is complete. One of the steps is that given $$||x_m - x_n||_\infty = \sup_{i\in \mathbb{N}} | {x_m}_i - {x_n}_i| \leq \epsilon,$$ then for each $I\in ...
4
votes
1answer
59 views

Convergence in $C(X)$ is uniform convergence.

I read this the convergence in $C(X)$ is uniform convergence. Where $X$ is compact hausdorff topological space and $$C(X)=\{f:X\to\mathbb{C}\;\mid \; f\ \text{is continuous}\}$$ And ...
1
vote
1answer
32 views

Prove this inclusion: $\bigcup_{k<p}\ell^k\subsetneq\ell^p$

Let $1<p<\infty$. I have to prove that $$ \bigcup_{k<p}\ell^k\subsetneq\ell^p. $$ I am not able to find a counterexample to prove the inequality.
1
vote
0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
2
votes
1answer
56 views

Show $T: C([0,1]) \rightarrow C([0,1])$ is compact

Consider $T: C([0,1]) \rightarrow C([0,1])$ defined by $$(Tf)(t) := \int_0^1 \kappa_t(s)f(s)ds,$$ where $\kappa:[0,1]^2 \rightarrow \mathbb{R}$ satisfies the following properties: for all $t\in ...
0
votes
1answer
32 views

Essential support vs. classical support for a continuous function

The essential support of a function $f:\Bbb R^n\rightarrow \Bbb R$ is defined in the following way: Let's denote $\mathcal A_f=\{\omega \subset \Bbb R^n: \omega \quad \text{open}, \quad f(x)=0\quad ...
1
vote
1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
2
votes
2answers
50 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
4
votes
2answers
98 views

Fubini's theorem and $\sigma$-finiteness?

I'm reviewing my analysis notes, and I am really confused about what is meant by $\sigma$-finiteness being a hidden hypothesis of Fubini's theorem. Here is Fubini's theorem as was stated to me: ...
5
votes
1answer
66 views

Show that a subspace of l2 is not complete

I would like to know if this exercise is correct. Let $\Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}$. Show that $(\Bbb R^\infty, \| ...
2
votes
1answer
56 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in ...
0
votes
1answer
19 views

Properties of compactly supported continuous function

Let $a,b,c \in R$ Let D = {$(x_1,x_2,x_3): x_1^2 + x_2^2 +x_3^2 \leq 1 $}. Let E = {$(x_1,x_2,x_3): \frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} + \frac{x_3^2}{c^2} \leq 1 $} and ...
1
vote
0answers
21 views

Show P is linear if it is convex and positive homogenious functional

it might be too simple but I couldnt show the second part L linear real space $P:L\rightarrow \Bbb R$ is called positive homogenious functional if for every $x\in L$ and $\alpha\ge 0$ , $P(\alpha ...
3
votes
1answer
46 views

Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...