Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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110 views

Sobolev Spaces and Derivative

I need help on the problem 8.9 at page 238 of the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis. Set $I=(0,1)$. Let $u \in W^{2,p}(I)$ with ...
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94 views

Why the space of probability measures is a subset of the measure space

Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R ...
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116 views

Composition of Partial Isometries

Let $H$ be a complex Hilbert space and $S,T \in B(H)$ partial isometries. Then $S T$ is a partial isometrie, if and only if $T^*(\ker(S)) \subseteq \ker(ST)$. Edit: My attempts so far: ...
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1answer
48 views

Baire Category for monotonic sequence

Let X be a non-empty complete metric space and let $\{f_n:X\to \Bbb R\}^\infty_{n=1}$ be a sequence of continuous functions with the following property: for each $c\in X$, there exists an integer ...
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1answer
95 views

Counterexample for Palais-Smale condition

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when ...
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1answer
65 views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases} ...
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67 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 ...
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165 views

Riesz Lemma for reflexive spaces

I know the proof of Riesz Lemma: Let $Y$ be a closed (proper) subspace of a normed space $X$. Let $\varepsilon >0$. Then it exists an element $x \in X$ such that $||x||=1$ and $d(x, Y) \geq ...
2
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1answer
70 views

Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
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35 views

Closure of a von Neumann bounded set

Let $V$ be a topological vector space and $B \subseteq V$ bounded. Then the closure $\overline{B}$ is bounded. This appears on the Wikipedia page ...
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85 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...
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1answer
228 views

Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)

I have two specific questions. The first is the result I actually need, and the second would let me prove it. EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third ...
2
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2answers
78 views

Frechet differentiable implies reflexive?

Note: The question has been cross-posted (and answered) on MathOverflow here. Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive?
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121 views

Every weak convergent sequence is bounded

Theorem: Every weakly convergent sequence in X is bounded. Let $\{x_n\}$ be a weakly convergent sequence in X. Let $T_n \in X^{**}$ be defined by $T_n(\ell) = \ell(x_n)$ for all $\ell \in X^*$. Fix ...
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1answer
50 views

Show that the limit is zero

I have to show that if $(x_n)$ is weakly convergent in $X$ then for any $a>1$ $$\lim_{n\rightarrow\infty}\frac{\|x_1+\dots + x_n\|}{n^a}=0$$ My attempt: If $(x_n)$ is weakly convergent, then it ...
2
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1answer
67 views

Density of $C^\infty(\overline{\Omega})$ in $L^2(\Omega)$: can we find a bounded sequence approximating $a \in L^2(\Omega)$

Let $a \in L^2(\Omega)$ (bounded $\Omega$) with $0 \leq a(x) \leq C$ a.e. We know $C^\infty(\overline{\Omega})$ is dense in $L^2(\Omega)$, so there exist smooth functions $a_n \to a$ in $L^2$. But ...
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166 views

Cayley Transform: well defined?

Why is the Cayley backtransformation well-defined: $$A_U:=\imath(1+U)(1-U)^{-1}$$ In general $1-U$ is not invertible for example for $U=1$.
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1answer
66 views

Two questions about a proof of the compactness of an operator

There are a few things that I don't understand about a proof and I'd appreciate any help. The theorem and its proof are the following: (1) Is the equality $$ \|v(\tau) -v(\tau_j)\| = \max_{1 \le ...
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1answer
207 views

Projection onto finite-dimensional subspace of $L^p$

Let $a_i$ be a basis of $L^p(\Omega)$ and consider $A_n = \text{span}\{a_1, ..., a_n\}$. Take an element $f \in L^p$. We want to define a projection onto the finite-dimensional subspace $A_n$. How do ...
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1answer
77 views

Why $(X,d)$ is a complete $\mathbb{R}$-tree?

Definition. An $\mathbb{R}$-tree is a metric space $(X,d)$ such that there is a unique geodesic segment (denoted $[x,y]$) joining each pair of points $x,y \in X$; if $[x,y] \cap [y,z] = ...
2
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1answer
88 views

Uniform convergence of $f_n(x)=nx^n(1-x)$ for $x \in [0,1]$?

I want to decide whether or not $f_n(x)=nx^n(1-x)$ is uniformly convergent or not. I have shown that $\lim_{n\to\infty} f_n(x)=0$ for $x \in [0,1]$. Now $f_n(0)=f_n(1)=0$. And in $(0,1)$, we have ...
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82 views

question concerning weak star convergence.

Given: X seperable Banach space. X' its dual. We have $M\subset X'$ a closed unit ball in X'. Choose a sequence $(x_{n}$) of nonzero elements in X which is dense in X. Define ...
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2answers
138 views

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ ...
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63 views

Second derivative test inconclusive, all derivatives are 0, moving critical point to origin, no result?

Here is a function $f(x,y)=x^4 + 6x^2y^2 + y^4 -4x^3 - 12xy^2 + 6x^2 + 6y^2 - 4x + 1$. I've happily proved that $(1,0)$ is a critical point for that function. Now I'd like to decide whether is it a ...
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1answer
44 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
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3answers
213 views

Is a function in $L^2$ which second derivative is in $L^2$ in $H^2$?

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary. Assume that $f\in L^2(\Omega)$ and $f^{\prime\prime}\in L^2(\Omega)$. Does one have $f\in H^2(\Omega)$? Useless comments: ...
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42 views

A soft question on the dimension of normed spaces

There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional. An example is of course the compactness of closed bounded sets. Another ...
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1answer
75 views

How to show that the sequence $(x^{(n)})$ weakly convergent in $l_p$, $1\le p\lt \infty$

How to show that the sequence $(x^{(n)})$ weakly convergent in $l_p$, $1\le p\lt \infty$. where $(x^{(n)})=(\underbrace{0,0,..0}_{n-1},1/n,1/(n+1),...,1/(2n),0,0,...)$ for $n\in\mathbb{N}$
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1answer
32 views

$\lim_{t\to 0}\sup_{s\in \mathbb{R}}|f(t+s)-f(s)|$ uniformly on $f$

Let $X$ be the space of all bounded uniformly continuous functions $f:\mathbb{R}\to \mathbb{R}$ equiped with the supremum norm $|f|_\infty$. We know that for each $f\in X$ we have $$\sup_{s\in ...
2
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1answer
71 views

Establishing an inequality between the $2^{nd}$ largest eigenvalue of $A$ and a related matrix.

Let $A$ be an irreducible, aperiodic matrix with non-negative entries, with $1 \in \ker(A - I)$, $w \in \ker(A^\top - I)$, $w_i > 0$ $\forall i$. Define $W = \text{diag}(w)$. I am studying the ...
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1answer
73 views

Showing that the set of functions in $C(I)$ which are monotone on some nontrivial subinterval of $I$ is of first category in $C(I)$.

Let $I = [0,1]$ and let $C(I)$ be the metric space of continuous functions on $I$ with the $L^{\infty}$ norm. I am trying to show that the set of functions in $C(I)$ which are monotone on some ...
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1answer
40 views

Inequality $||f-g|| < \epsilon \Rightarrow |E[f] - E[g]| < \epsilon$

Let $C(X)$ be the space of continuous bounded functions on some metric space $(X,d)$. Can it be shown that if $||f-g||_\infty < \epsilon$ if follows that $| \int f \, \text{d}P - \int g \, ...
2
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1answer
79 views

“Duality” for weak $L^p$ spaces

Let $1<p<\infty$. Denote by $L^{p,\infty}$ the weak $L^p$ space in $\mathbb{R}^n$ and let $f\in L^{p,\infty}$ where we define the weak $L^p$ quasinorm as $$\|f\|_{p,\infty} = \sup_{\lambda ...
2
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1answer
105 views

Proving functions are uniformly continuous help!

a) Prove that $f(x)=x^{1/4}$ is uniformly continuous on $[0,\infty)$. Show that this method can be extended inductively to any $f(x)=x^{1/p}$ for any $p=2^n$ b) Prove directly from the definition ...
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1answer
43 views

A linearly independent set about approximate units

Let $B$ be a C*-algebra and $\{b_{i}\}_{i=1}^{n}\subset B$ be linearly independent. If we take $\{f_{k}\}\subset B$ which is approximate units, then can we find a large $k$, such that $\{b_{1}f_{k}, ...
2
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1answer
122 views

Help explain why the proof works - Gradient Estimate Using the Maximum Principle

I found the following proposition and proof in a textbook for one of my classes, and the end of the proof seems a little "unfinished" to me. In other words, as the author left it, I'm not entirely ...
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1answer
125 views

the principle of uniform boundedness and $l^p$ space

If $1<p<\infty$ and $\{x_n\}\subset l^p$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in l^q$, $\frac{1}{p}+\frac{1}{q}=1$, iff $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ for every ...
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1answer
48 views

Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help
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87 views

When does $ f'_{n}(x) \to g(x) =1$ imply $f'(x) =1 $

I considered the following: $f_{n}(x) \in C^1(0,1)$ (class of continuously differentiable functions) and $f_{n} \to f:(0,1) \to \bf{R} $ with $f'_{n} \to g =1$. Does this imply that $f \in C^1(0,1)$ ...
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1answer
61 views

Convergence of function in $L^1$ space

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
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1answer
84 views

An exercise about projections on Hilbert space

Let $H$ be a Hilbert space with an orthonormal basis $\{v_{n}\}_{n=1}^{\infty}$. The C$^{*}$-algebra $K$, the set of all compact operators on $H$, is a non-unital C$^{*}$-algebra. Let $p_{n}$ be the ...
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1answer
36 views

Approximating a mean on $L^{\infty}(G)$ by a norm $1$ non-negative $f\in L^{1}(G)$

Let $G$ be a locally compact group. A mean $M$ on $L^{\infty}(G)$ is a continuous linear functional on $L^{\infty}(G)$ such that $1 = \langle 1 , M\rangle = \|M\|$. My Exercise: Show that the set ...
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1answer
136 views

Properties of this ODE

Given the ODE $$ \{e^{\alpha\sin\theta}\Psi'(\theta)\}'+ e^{\alpha\sin\theta}\beta\cos^{2}\theta \Psi(\theta) = \lambda e^{\alpha\sin\theta}\Psi(\theta). $$ where $\theta \in ...
2
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1answer
82 views

Convergence Radius: Non-Analyticity

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
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1answer
36 views

$\|f'(x)\|_{L^p} \le C \|f(x)\|_{L^p}^{1/2} \|f''(x)\|_{L^p}^{1/2}$ for smooth $f$ with compact support

I'm trying to prove the following Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function supported on $[a, b]$ where $-\infty < a < b < \infty$. $2 \le p < \infty$. Then $$ ...
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1answer
77 views

Unit ball on $B(H)$ and the weak -topology

i have the following problem: i can show that the map $d:B(H)\times B(H)\rightarrow \mathbb{R}$ (with $H$ a separable Hilbert space and $(e_n)_{n\geq1}$) given by: ...
2
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1answer
106 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
2
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1answer
50 views

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$. I'm trying to show wheter S can be written as $\cup_{n=1}^\infty S_n$ where $S_n\in X$ and $Int\bar{S_n}=\emptyset$ I tried a ...
2
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1answer
486 views

Equi integrability and weak convergence of measures

Let $f_n$ be a sequence of functions in $L^1(K, m ; \mathbb{C})$, $K$ metric compact and $m$ a Radon measure on $K$. Assume that $\| f_n \|_1 \leq 1$. From what I understand, there is a subsequence ...
2
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1answer
61 views

How to show: $\exists$ L such that $X\neq L\oplus L^{\perp}$

$X$ : inner product space $L$ : closed subspace of $X$ How to show: $\exists$ L such that $X\neq L\oplus L^\perp$ let $X= (\text{the set of all finite sequences}, \|\cdot\|_2 )$ $L=\lbrace x\in ...