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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Tracial states on $C(X,M_n(\Bbb{C})$

Let $X$ be a compact Hausdorff space. $M_n(C(X))$ is identified with $C(X,M_n(\Bbb{C}))$ Show that any tracial state, $\tau:M_n(C(X))\to \Bbb{C}$ is of the form: $\tau(a)=\int_X Tr(a(x))d\mu $ ...
7
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2answers
724 views

Monotone Convergence Theorem for Riemann Integrable functions

I'm having a really hard time proving this statement (this is not homework): If $f_{n} : [0,1] \rightarrow \mathbb{R}$ is a Riemann integrable function for all $n \in \mathbb{N}$, and $0 \leq f_{n + ...
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40 views
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convergence in distribution in Banach spaces

We let $X$ be a compact metric space and consider $C(X)$ to be the space of all continuous functions on $X$. The dual space of $C(X)$ can be seen as the set of all signed borel measure on $X$. My ...
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57 views
+50

$||\hat{f} ||_{\infty} = \lim _ {n \rightarrow \infty} (||f^{(n)}||_1)^{1/n}$

Let $f \in L^2 \cap L^1$ on the Real line, and define $f^{(n)}$ to be the $n$-fold convolution $f \circ f ... \circ f $. I want to show that $||\hat{f} ||_{\infty} = \lim _ {n \rightarrow \infty} (||...
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14 views

Long time asymptotic of Fokker–Planck equation $\; \partial_tu-\nabla\!\!\cdot\!\left(\nabla u+xu\right) = 0$

Is it true that given a solution to the Fokker–Planck equation $$\partial_tu-\nabla\!\!\cdot\!\left(\nabla u+x\hspace{0.2ex}u\right) = 0,$$ then we have $$ \left\|\frac{u-\rho}{\rho}\right\|_{\...
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0answers
12 views

Trace Theorem and Neumann boundary.

I've been studying Trace Theorem. From PDE Evans, we have THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator $$T : W^{1,p}(U) \...
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1answer
14 views

Increment of a functional and Taylor's theorem

In Gelfand and Fomin (Calculus of Variations) at page 14 they derive a formula for a certain variation. My problem is just one part of that derivation. $$\Delta J= J[y+h]-J[y]=\int_{a}^{b} [F(x,y+h,y'...
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1answer
17 views

Motivation for condition of Normed Algebra

Is there any particular motivation for the defining Normed Algebra with the condition $\|xy\| \leq \|x\|\|y\|$. Is there any Geometrical view of this condition?
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0answers
30 views

A bound for a solution of a PDE

Let $u(t,x):\mathbb{R_+}\times\mathbb{R}\rightarrow\mathbb{R}$ be a very smooth function, which satisfies the equation: $$\dfrac{\partial u}{\partial t}+f(x,u)\dfrac{\partial u}{\partial x}=g(x,u),$$ ...
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1answer
13 views

Strongly measurable implies Borel measurable in separable space

Let $(M,\mu)$ be a measure space, $X$ be a Banach space, $f: M \to X$ be a function. $f$ is said to be strongly measurable if there is a sequence of simple functions $\{f_n\}\to f$ pointwisely a.e.. $...
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0answers
37 views

Quotient map and compactness [on hold]

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. Let $\pi :X\longrightarrow \frac{X}{Y}$ be the quotient map. If $\pi T\arrowvert _Y$ be a compact operator, can we say there exists a ...
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0answers
29 views

Existence of separating hyperplane after injective mapping of compact set

Let $f: X \to Y$ be an injective continuous function where $X \subset \mathbb{R}^m$ is a nonempty compact set, and $Y \subset \mathbb{R}^n$. Let $\mathbf{y}^* = f\left(\mathbf{x}^*\right)$. If $E \...
2
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2answers
24 views

About a partial converse to the Banach-Steinhaus Theorem

I've been reading the GTM text Topics in Banach Space Theory by Albaic and Kalton. In the appendix, it states the following partial converse to the Banach-Steinhaus theorem: Let $\{S_n\}$ be a ...
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2answers
16 views

If $A$ is a compact operator, is $\overline{A(B_1(0))}$ finite dimensional?

Let $A$ be an operator. An operator is called compact iff $\overline{A(B_1(0))}$ is compact. A normed space is finite iff $\overline{B_1(0)}$ is compact. Let $X$ be a Banach space and $Y$ a Hilbert ...
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0answers
10 views

A $*$-closed algebra of compact operators is completely reducible

In page 13 of Lang's $SL_2$ there is a proof that for a $*$-closed algebra $\mathscr A$ of compact operators on a Hilbert space $H$, $H$ is completely reducible. The proof follows by taking the ...
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1answer
16 views

If $H$ is a Hilbert space, $U≤H$ is closed and $E≤U^⊥$ such that $x∈H$ with $x⊥_H E$ implies $x∈U$, then $U^⊥=\overline E^{\langle\;⋅\;,\;⋅\;\rangle}$

Let $\left(H,\langle\;\cdot\;,\;\cdot\;\rangle\right)$ be a separable Hilbert space $U$ be a closed subspace of $H$ $E$ be a subspace of $$U^\perp:=\left\{x\in H:\langle x,u\rangle=0\text{ for all }...
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0answers
25 views

Convergence of measures in total variation sense

Suppose we have measures that defined over the set $\{0,1,2,..,C\}$. Let $\{\mathbb{P}_{n,m}\}$ be a sequence of measures. Suppose that for fixed $n$, $\mathbb{P}_{n,m}$ converges to $\mathbb{P}_n$ ...
3
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0answers
37 views

How to solve a Volterra integral equation of the second kind

I have the following equation \begin{equation} F(\theta) + (c)^{\frac{1}{c}-1}\sqrt{\frac{c}{2\pi}}\int\limits_0^\theta \frac{\theta^{\frac{1}{c}} - \tau^{\frac{1}{c}}}{(\theta - \tau)^{3/2}}\exp{\...
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0answers
27 views

I don't understand De Rham's theorem about the gradient of a distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $$\mathfrak D(\Omega):=\left\{\Phi\in\mathcal D(\Omega)^d:\nabla\cdot\Phi=0\right\}$$ In a ...
2
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1answer
32 views

Is this Adjoint Operator Self Adjoint?

I'm helping some students study for their qualifying exam, and I wanted to double check my interpretation of a question. Suppose we define the operator $L$ on $H=L^2\left([0,\infty)\right)$ so that ...
9
votes
2answers
2k views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
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0answers
27 views

$\nabla u \in L²(B_1(0)) \Rightarrow u \in H^1(B_1(0))$ [on hold]

how do I show the following: $\nabla u \in L²(B_1(0)) \Rightarrow u \in H^1(B_1(0))$. Thanks for your answers.
2
votes
0answers
52 views

Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0, \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}^{N}$ is open ...
2
votes
0answers
34 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
0
votes
0answers
30 views

Hahn-Banach Theorem in metric space

Hahn-Banach Theorem is important theorem on vector space. is Hahn-Banach Theorem in metric space? especially for Hadamard space?
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0answers
14 views

Characterization of the Gradient of a Distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ (without topology) $u:\mathcal D(\Omega)\to\mathbb R$ is called distribution on $\Omega$ $:\...
5
votes
1answer
92 views

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
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0answers
22 views

Something is wrong with this argument (Lorentz and Rosenthal-Woo sequence spaces)

Fix once and for all $1<p<\infty$. Throughout, $w=(w_n)_{n=1}^\infty$ will denote a sequence of positive real weights satisfying \begin{equation}1=w_1\geq w_2\geq w_3\geq\cdots>0\;\;\;\text{ ...
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1answer
39 views

Definition of the space $H^s(\mathbb{R}^n)$

The following is the definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis: Here a regular distribution is a tempered distribution $T_f$ such that it is given by $$ T_f(\varphi)=\...
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1answer
27 views

Dimension of $C^n$, space of all $n$-differentiable functions?

What is the proper classification for the (infinite) dimensionality of $C^n$, the space of all functions (defined on $\mathbb{R}^m$, $m\in\mathbb{N}$) with continuous derivatives from order 0 to $n$? ...
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0answers
28 views

Weak-* compactness of probability measures on compact, non-Hausdorff space

Let $X$ be a compact topological space. How would I prove that the space of probability measures $\Delta(X)$ is weak-* compact? Without assuming $X$ being Hausdorff I can't apply to the usual Reisz ...
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votes
0answers
12 views

Characterization of a set occurring in the Helmholtz-Hodge decomposition

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 2$ Each $f\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal ...
3
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0answers
38 views

Proof of the Helmholtz-Hodge decomposition

Let $\Omega\subseteq\mathbb R^3$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$...
1
vote
1answer
45 views

Show the following space is Banach

Show that $(X, \| \cdot\|_\star)$ is a Banach space, where $X$ is the following linear space: $$ X = \left\{ f : \mathbb{N}_{\ge 1} \to \mathbb{R} \mid \|f\|_\star := \left( \sum_{k=1}^\infty (k+1) |f(...
2
votes
0answers
34 views

Resolvent Inequality

Let $H$ be a Hermitian matrix and $h$ some vector of the same length. The resolvent of $H$ at $z\in\mathbb C$ shall be denoted by $$G(z):=(H-z\cdot1)^{-1}.$$ Is it true that $$\frac{(\Im z)\left(1+\...
3
votes
1answer
42 views

All nonzero singular values of $A$ are equal to $1$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
0
votes
0answers
20 views

Proof of the Sobolev Space chain rule from Kesavan's Book

I put chain rule on the title because that's what I think they are asking here: This is taken from Kesavan's Functional Analysis book, exercise 2.9 Suppose $\Omega_1 $ , $\Omega_2 $ are bounded open ...
5
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1answer
62 views

Is $\Delta C_c^\infty$ a dense subset of $L^p(\mathbb{R}^d)$?

I'm struggling to obtain some density result. It is well known that $C^\infty_c(\mathbb{R}^d)$ is dense in $L^p (\mathbb{R}^d)$ for $1\leq p<\infty$. It is well known that for $\lambda>0$, $(\...
3
votes
1answer
71 views

What is the mathematical meaning of a quantum operator?

(Context: I am learning functional analysis using the book by Erwin Kreyszig "Introductory functional analysis with applications." The last chapter is dedicated to the applications of functional ...
2
votes
1answer
30 views

Completion of a normed space using an isometry

I'm practicing some old exams for my functional analysis exam tomorrow, and i'm having trouble with the following: Let $X$ be a reflexive Banach space and let $Y$ be a normed space. Assume there ...
1
vote
1answer
112 views

States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
3
votes
2answers
41 views

Application Banach-Alaoglu Theorem

When reading about Banach-Alaoglu Theorem on Wikipedia, I read the following assertion: '' Let $f_n$ be a bounded sequence of functions in $L^p$. Then there exists a subsequence $f_{n_k}$ and an $f\...
5
votes
2answers
137 views

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ||x||\...
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votes
1answer
19 views

Span of Hilbert base

Let $\{e_j\}$ be an (infinite) Hilbert base of a Hilbert space $H$. Is the subspace $U=span_{\mathbb{C}}\{e_j|j\ge 0\}$ again a Hilbert Space? Thanks
3
votes
1answer
40 views

Show that $A\varphi_j=\left<A\varphi_j,\varphi_j\right>\varphi_j$ and $A^*A\varphi_j=s_j(A)^2\varphi_j$ for all $j$

Let $A$ be a bounded linear (compact) operator acting on a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_j,\...
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vote
1answer
34 views

Vainberg Theorem in measure theory

In a lecture notes about Variational Methdos, I found the following theorem: THEOREM: Let $(f_n)$ a sequence in $L^{p}(\Omega)$ and $f \in L^{p}(\Omega)$, such that $f_{n} \rightarrow f$ in $L^{p}(\...
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0answers
22 views

lower $\ell_q$ estimate for monotone positive weighted sequences

Fix $1\leq p<\infty$. Conjecture 1. There exists $q\in(p,\infty)$ and $C\in[1,\infty)$ such that for all $m\in\mathbb{N}$ and all positive nonincreasing sequences\begin{equation}a_1\geq a_2\geq\...
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1answer
22 views

Reference: Gaussianity of linear functional of Gaussian process

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets, If you take $X$ in $C([a,b])$ [i.e., $X$...
1
vote
2answers
69 views

Does this function have an inverse?

$$f(x)=2^{2x}-2^{x+1}$$ Is this function one-to-one and does it have an inverse in $\Bbb{R}$ ? And if the answer is yes , what is the formula for it ? I tried many ways like plotting it but I can't ...
-1
votes
4answers
66 views