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1 vote
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Constructing seminorm from norm

We can describe these examples by composing with a suitable linear operator. It's easier to describe the $L^1[0, 1]$ example first: we have $$\int_0^k |f(x)| \, dx = \int_0^1 |f(x) 1_{[0, k]}(x)| \, ...
0 votes

Completeness of the sum of two $L^p $ spaces

Suppose $f_i$ is a Cauchy sequence in $L^{p_0}+L^{p_1}$. By definition, for every $\epsilon > 0$, there is an $N$, such that $\|f_n - f_m \|_{L^{p_0}+L^{p_1}} < \epsilon$ for any $n,m \geq N$. ...
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Let $X$ be compact metric space and $\mu_n$ be sequence of complex measures on it. Prove the following two representations of $C(X)$ is equivalent.

Your $U$ is onto because your $\mu_n$'s are mutually singular: let $\{X_n\}$ be a partition of $X$ by Borel sets such that $\mu_n=0$ outside $X_n$. Then, for any $g=\{g_n\}\in H$, wlog (up to equality ...
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2 votes

Second Derivative Test in Banach Spaces

daw gave a nice answer/counterexample. I want to add a more geometric interpretation. There is a version of Morse Lemma in the Hilbert setting, which asserts that if $x_0$ is a non-degenerate ...
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6 votes

Second Derivative Test in Banach Spaces

The theorem is not true. Here is a counterexample: Take $E=l^2$, $D=E$, and $$ f(x) = \sum_k \frac1k x_k^2 -x_k^3. $$ Then $$ f'(x)v = \sum_k (\frac2k x_k -3x_k^2)v_k, $$ $$ f''(x)(v,v) = \sum_k (\...
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2 votes
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Inverse of a bounded operator on a Hilbert space

Consider $H = \ell_2(\mathbb{N})$, and let $T$ be the shift operator $T(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$. Then $\Vert Tx \Vert = \Vert x \Vert$ for all $x$, but $T$ is not surjective so it ...
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1 vote
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Let A and B be infinitesimal generators of two semigroups in space X. Show an example where if D(A) ⊂ D(B), but D(A) different D(B)

For $L^2(1,\infty)$ let $A$ and $B$ act by multiplying by $-x^2$ and $-x$ respectively with domains $$D(A)=\{ f\in L^2\,:\, x^2f\in L^2\}, \quad D(B)=\{ f\in L^2\,:\, xf\in L^2\}$$ Then $D(A)\...
0 votes

Positive homogeneous norm

I think they will be equivalent. Let me explain my reasoning, which also works in the more general case of two functions positively homogeneous and subadditive functions $p$, $q$ satisfying $p(x)=0$ ...
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Coercivity of an integral operator in $L^2$-norm

The claim is false. Here is a counterexample. For each $\epsilon>0$, consider $k^\epsilon_t=(1+\epsilon-t)^{-1}$ for all $t\in (0,1)$. Note that $k^\epsilon \in L^2(0,1)$, with $\int_0^1 (k^\...
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1 vote
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Approximation of piecwise linear function where $f_n(x)=0$

There are a few ways to resolve this, but generally your goal will be to define a sequence of functions $f_n \in C^\infty$ (or $C^3$, or whatever) such that $f_n = f$ outside $(-1/n, 1/n)$ and there ...
1 vote
Accepted

Are Fourier transforms over $L^1(\mathbb{R})$ themselves in $L^1(\mathbb{R})$?

Consider the Fourier-transformation on $L^1(\mathbb R)$ given by $$F^1 \colon L^1(\mathbb R) \to C_0(\mathbb R), \quad f \mapsto \widehat f.$$ The fact that $F^1$ maps into $C_0(\mathbb R)$ is usually ...
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4 votes
Accepted

Inequality with norm in Space $L^2(\Omega)$

Let $A\subset \Omega$ satisfy $0<|A|<1.$ For $u= |A|^{-1/4}1\hspace{-2.5pt}{\rm I}_A$ we have $$\|u^2\|_2=1,\quad \|u\|_2=|A|^{1/4}<1$$ Hence for any constant $k>0$ the inequality does ...
2 votes
Accepted

The normed space of functions such that $x \mapsto \frac{f(x)}{x}$ is integrable

Let $Uf=xf.$ Then $U$ is the isometry from $L^1$ into $X.$ Moreover $U$ is surjective. Indeed if $g\in X$ then $f:=x^{-1}g\in L^1 $ and $Uf=g.$ As $L^1$ is complete so is $X$ as it is isometric with $...
0 votes

Prove that $\sum_{n=1}^\infty (-1)^n \frac1n$ is convergent but not unconditionally convergent

$\lim_{N\rightarrow\infty}\sum_{n=1}^{N}(-1)^n\frac{1}{n}$ does exists because $$ (-1)^n\frac{1}{n}+(-1)^{n+1}\frac{1}{n+1} \\ = (-1)^n\left[\frac{1}{n}-\frac{1}{n+1}\right] \\ = (-1)^n\...
3 votes

Inequality with norm in Space $L^2(\Omega)$

By Cauchy-Schwarz inequality you obtain $$ \| v \|_{L^2(\Omega)}^2 = \int_\Omega v^2 dx \leq \|1\|_{L^2(\Omega)}\|v^2\|_{L^2(\Omega)} = |\Omega|^{1/2} \|v^2\|_{L^2(\Omega)}, $$ hence the inequality ...
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0 votes

Prove that $\sum_{n=1}^\infty (-1)^n \frac1n$ is convergent but not unconditionally convergent

For the convergence, you can use the Leibnitz Test. We consider $a_n=\frac{1}{n}$ . The $(a_n)$ is a decreasing sequence, and $\lim_{n\to \infty}a_n=0.$ Thus, we have by Leibnitz Test that the series $...
0 votes

Linear dimension of banach spaces

See the article H. Elton Lacey, The Hamel Dimension of any Infinite Dimensional Separable Banach Space is $c$, Amer. Math. Mon. 80 (1973), 298, where a simple proof is given of the fact that every ...
1 vote

Upper Bound for Operator Norm in Marcinkiewicz Interpolation Theorem

Matt's calculations are right. To get the desired result, we only need to apply the Young's inequaly, i.e., $$ A^{\theta}B^{1-\theta}\le \theta A+(1-\theta)B, $$ when $A,B>0$ and $\theta\in (0,1)$. ...
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1 vote
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Prove that $\limsup\limits_{r\rightarrow 1} rf(r)|f'(r)|=0$

I had previously posted an incorrect proof of (1.) and (2.) - thanks to @ryszard-szwarc for pointing out the error! In fact, nether (1.) nor (2.) hold in general. We will first construct a function $g ...
1 vote
Accepted

Complex sequence space is complete (with a certain metric)

Yes, it is. Just note that $\frac {|a|}{1+|a|} <\epsilon$ iff $|a|<\frac {\epsilon} {1-\epsilon}$. (Without loss of genearlity assume that $0 <\epsilon <\frac 1 2 $ so that $\frac {\...
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3 votes
Accepted

Elliptic partial differential equations with robin boundary condition and domain of fractional power of Robin Laplacian operator

I] First the equivalence of norms. We set, \begin{align*} \lVert u\rVert_{\mathrm{H}^1(\Omega)} := \left( \lVert u\rVert_{\mathrm{L}^2(\Omega)}^2+\lVert \nabla u\rVert_{\mathrm{L}^2(\Omega)}^2\right)^{...
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0 votes
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Notation for multivariate Lebesgue space

No: $L^p(\Omega)$ with $\Omega\subset \mathbb R^d$ consists of (equivalence classes) of functions from a subset of $\mathbb R^d$ to $\mathbb R$, while $L^p(\Omega)^d$ with $\Omega\subset \mathbb R$ ...
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1 vote
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Numerical range and essential spectrum

$\def\e{\varepsilon}$ Yes. One characterization of the essential spectrum is that $E_A(\alpha-\e,\alpha+\e)$ is infinite for any $\e>0$. Then we can choose an orthonormal sequence $\{e_n\}$ with $...
6 votes
Accepted

Why does $\langle (U^*U - I)v, v\rangle = 0 \ \forall v$ imply that $U^*U = I$, from the polarization identity?

Let $M = U^*U - I$. We are given that $\langle Mv,v \rangle = 0$ holds for all $v \in \mathcal H$. On the other hand, the polarization identity allows us to express $\langle Mx,y \rangle$ in terms of ...
1 vote
Accepted

Maximization problem in order to find the norm of a linear functional.

Here's a quick trick: from general Hilbert space theory, if $H$ is a Hilbert space and $\xi\in H$ is a vector, then $\phi:H\to\mathbb{R}$ given by $\phi(x)=\langle x,\xi\rangle$ is a bounded linear ...
1 vote
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Proof of the Meyers–Serrin theorem (the "$H=W$" theorem)

Let's start from $S\subseteq W^{m,p}(\Omega)$. Consider the closure $\bar{S}$ of $S$ with respect to the topology on $W^{m,p}(\Omega)$ induced by its norm. Claim 1: $\bar{S}$ is a linear subspace of $...
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1 vote
Accepted

A bound for a vaguely convergent sequence of measures

$X=(0,1), \mu=n\delta_{1/n}, \mu=0$ is a counter-example.
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1 vote
Accepted

Orthonormal sequence $L^2[0,1]$ (Brezis 5.30)

The Fourier series for $\chi_{[0,t]}(s)p(s)$ is $$ \sum_{n=1}^{\infty}\int_0^1\chi_{[0,t]}(s')p(s')e_n(s')ds'\, e_n(s) \sim \chi_{[0,t]}(s)p(s) \\ \sum_{n=1}^{\infty}\int_0^tp(s')e_n(s')ds'...
2 votes

unbounded inverse of injective linear operator

You do not need any sort of topology to prove that the injectivity of a linear map only depends on its kernel: ($\implies$) If $T$ is injective and $Tx=0$, then $Tx=T0$ so $x=0$ by the definition of ...
0 votes

If $\mu(A\triangle B)=0$ then $\mu(A)=\mu(B)$

You can use awnser of Sayan Dutta above + the following below \begin{align*} \mu(A)=\mu((A\cap B)\cup (A\cap B^c)) &=\mu(A\cap B)+\mu(A\cap B^c)\\ &=\mu(A\cap B)+\mu(A\setminus B)\\ &=\mu(...
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-1 votes

I don't understand which norm exactly should be calculated, the norm of T or T(f)(x)? please give an example

First I give the definition of a norm. $\left\|T \right\|=\sup\left\|T(f) \right\|$ over all $f$ such that $\left\|f \right\|=1$ The a) is not correctly defined. Please make a correction, because $T$ ...
-1 votes

I don't understand which norm exactly should be calculated, the norm of T or T(f)(x)? please give an example

$C[0,1]$ is a Banach space under the sup norm. so i think you are supposed to compute: $$||T||=\sup_f\left\{\frac{|Tf(x)|}{|f(x)|}\right\}$$ since $f\in C[0,1]$, $f$ is bounded.
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1 vote
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Rudin's RCA: $2.14$ theorem

$\newcommand{\d}{\,\mathrm{d}}$Rudin writes: "$\mu_1$ and $\mu_2$ are measures for which the theorem holds". In particular the assumption/assertion $(1)$ holds for both $\mu_{1,2}$. That is: ...
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0 votes

Measure preserving transformation and ergodic implies unbounded return time for some measurable set

In case someone is interested, I wanted to prove the stronger version of this exercise where $T$ need not be invertible (see exercise 9 of page 56 in Petersen's Ergodic Theory), and interpreting "...
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0 votes

If $f$ is continuous on $\mathbb{R}$ and $f\to 0$ when $x\to \pm \infty$, is $f\in L^p(\mathbb{R})$ for some $1\leqslant p <\infty$?

$\newcommand{\d}{\,\mathrm{d}}$Your suspect counter-example is indeed a counter-example. I assume you had difficulty proving this, so I’ll fill in the details. Let $f:\Bbb R\to\Bbb R$ be given by $(\...
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1 vote
Accepted

How can I prove that $\sup\{\Phi(f):f\in \bar B\}=1$ if $\bar B=\{f\in C\left([0,1]\right): d(f,0)\leq 1\}$?

Step 1 is correct. For Step 2 you can consider functions in $\bar B$ with $f(1) = 1$ and $\int_0^1 |f(t)| dt$ arbitrarily small, for example $$ f_n(x) = \begin{cases} 0 & 0 \le x \le 1-1/n \\ n ...
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0 votes
Accepted

Is $A\setminus \{x_1,\dots,x_n\}$ an open set if $A$ is open?

Yes, that is absolutely correct. The intuition is that if $a\in A\setminus\{x_1,\dots,x_n\}$, then $a$ is an interior point of $A$, i.e. you can draw an open ball centred around $a$ which is wholly ...
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1 vote
Accepted

Approximating a normal operator by invertible normal operators

Given $k\in\mathbb N$, there exists $f_k:\sigma(N)\to\mathbb C$, simple, with $\|f_k-\operatorname{id}\|<\frac1k$. We can write $$ f_k=\sum_{j=1}^{m_k}\alpha_{k,j}\,1_{E_{k,j}}, $$ where $\...
1 vote

Is there a symmetric real valued function on ${\mathbb R}^n$ that lies between sum and product?

As Hagen says in the comments, the strict inequality $f_n(x_1, \dots x_n) > \prod x_i$ is unsatisfiable at $(1, 1, \dots, 1)$. If you weaken this condition to a weak inequality then you can take $...
1 vote
Accepted

Lagrange multiplier $f(x,y)=x^2+2xy^2+y^2$. Show that $(-1,1)$ is the minimal point of $f$.

The solution (-1,1) is not a global min. The problem has no global min, it is unbounded to $-\infty$. To see that make the substitution $x=1-2y$ in the objective function to have now an unconstrained ...
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1 vote

Is the following operator positive?

The inequality does not hold even for two dimensional space. For $0<\delta<1$ and $ 0<a<b<{1\over 2}$ let $${\rho} =\begin{pmatrix} 1&0\\ 0&\delta\end{pmatrix},\quad M =\begin{...
1 vote
Accepted

Finding the spectrum of $(Tf)(t)=\max[0,\cos(t)]f(t)$ for $f\in L^2[-\pi,\pi]$

The point spectrum $\sigma_p(T)$ consists of all $\lambda$ for which there exists $f\in L^2[-\pi,\pi]\setminus\{0\}$ such that $Tf=\lambda f$. It is not hard to see that $0$ is the only value in the ...
2 votes

Given a derivative, say, as a tensor, is the corresponding linear approximation unique?

What are the hidden assumptions in this interpretation? There are no hidden assumptions. Here is how to do this calculation: if $A$ is an invertible matrix, then for $\varepsilon$ a sufficiently ...
1 vote
Accepted

Scaling properties of Hölder seminorms

Let $X$ be defined on $[0,1]$ by $X=X'\circ\varphi$ with $X'$ defined on $[s,t]$ and $\varphi:[0,1]\to[s,t],\;x\mapsto s+(t-s)x$. Then, $\varphi$ is increasing and onto, hence $$\begin{align}\|X'\|_{\...
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1 vote
Accepted

Direct sum of Hilbert modules

You can make $E$ into a $C^\ast$-module over $A\oplus B$ by considering the action $e(a,b)=ea$ and the new inner product $\langle e_1\mid e_2\rangle=(\langle e_1,e_2\rangle,0)$. If you do the same ...
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0 votes

Nonlinear function continuous but not bounded

I have two questions on the above example for $C[0,1]$ of orangeskid (apologies, if this is misplaced, I may change this into a new question, but I thought it would be easier to read here) (1) I ...
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0 votes
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Relatio between Frechet derivative and partial derivatives in $\mathbb R^n$

(We assume we are in $\mathbb{R^{n}}$). If all partial derivatives exist and are continuous, then is Frechet differentiable (since $\bigtriangledown f(x)$ exists and it is continuous) and also $C^{1}$....
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Example of a C^n Gateaux differentiable function which is not Frechet differentiable

If $P$ is continuously Gateaux differentiable on an open set then it is Frechet differentiable there, which can be proven using this mean value theorem. That is, apply the mean value theorem to $h \...
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2 votes
Accepted

Continuously differentiable functions is closed in $H^1(\Omega)$

This is not true. Take $\Omega=(-1,1)$, $y(x)=|x|$. Define $y_k(x):=\sqrt{y(x)^2 + \frac1k}$. Then $y_k \to y$ in $H^1$, which can be proven by dominated convergence. And $C^1$ is not a closed ...
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0 votes

Convergence of closed formula for Moyal product

This paper by Waldmann discusses precisely this type of question: https://arxiv.org/abs/1901.11327 Essentially there seem to be two interpretations (see section 2): One can view the power series as ...
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