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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10 views

Two characterizations for the interior and closure of a convex and absorbing set when the Minkowski functional is continuous

Let $(X,\|\;\|)$ be a normed vector space over $K$ and E$\subset X$ be convex and absorbing. Lets define $E_1=\{x\in X:p_E(x)<1\}$ and $E_2=\{x\in X: p_E(x)\le 1\}$. I want to prove that: $$ ...
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34 views

Why is $(\sqrt{P})^2=P$ where $P$ is a positive operator on a Hilbert space?

The following is a proposition regarding positive operators on a Hilbert space in Douglas's Banach Algebra Techniques in Operator Theory: Corollary 4.32 is as the following: I understand that the ...
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33 views

Theorem (Continuous compact function in compact $E\subset\mathbb{R}^n$)

Above all, I want to say I am Sorry for posting many questions. Theorem 1.15 (Continuous compact function in compact $E\subset\mathbb{R}^n$) Let $E$ be a compact set in $\mathbb{R}^n$ and $f$ be ...
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25 views

Prove generalisation of Sobolev embedding theorem using induction.

I am trying to prove the following; I am doing it by induction and the case $k=1$ is already done. So suppose the above is true for all integers less than or equal to k, first we want to show that ...
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28 views

approximate $C^1$ function by Holder function

I am trying to prove the following Lemma: Suppose $u$: $\mathcal R^N\to \mathbb R$ is $C^1$. Then for each $\epsilon>0$, there exists a $C^{1,\alpha}$ function $\tilde f$ such that $$ ...
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51 views

What is different between $\frac{1}{1+\lambda x}$ and $\exp{(-\lambda x)}$

I want to choose a function $f(x)$ which has properties: $f(x)$ closes to $0$ when $x$ goes to $+\infty$ . I have two option for that $f(x)=\frac{1}{1+\lambda x}$, where $\lambda$ is tuning ...
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16 views

If F is convex and lower semicontinuous in norm, then F is weakly lower semicontinuous

I have to prove the following statement: If $F$ is convex and lower semicontinuous in norm, then $F$ is weakly lower semicontinuous.
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40 views

Do compact convergence topology and w*-topology coincide on the Pontryagin dual group of a LCA group.

Given a locally compact group $G$, consider its Pontryagin dual $G^{'}$. Do compact convergence topology and weak*-topology on $G^{'}$ coincide?. When we talk about weak*-topology on $G^{'}$, it ...
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44 views

Products of functions is Hölder continous

I have $A$ maps $\Omega$ into the $n\times n$ real matrices, where $\Omega$ is a open bounded subset of $\mathbb{R}^N$. $C^{\alpha}(\Omega)$ denotes the set of functions that are Hölder continuous ...
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31 views

Unbounded linear operator with bounded restriction

Given that a linear operator $T:X\rightarrow Y$, where $X$ and $Y$ are both Banach spaces, $D$ a dense subspace of $X$, if we know that the restriction of $T$ to $D$, say, $S=T|_{D}$ is bounded, then ...
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23 views

The closure of a set as intersection of sum of two sets

Let $(X,\|\;\|)$ be a normed vector space over $K\;(\Bbb R\;\text{or }\Bbb C)$ and let $\;A\subset X$ then a I want to prove that: $$\bar A = \bigcap_{n\in\Bbb N} \big(A+B_{\frac{1}{n}}(0)\big)$$ So, ...
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35 views

Variational inequalities and their relation to weak formulations of PDEs

this topic is continuously boggling me as I go through material of a class on nonlinear PDEs. I will illustrate the broader issue with the following example. Consider the reaction diffusion equation ...
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38 views

Equality dimension between the space of bounded, convergent and convergent to zero sequences

Let: $$c(K)=\{x:\Bbb N\to K \mid \lim_{n\to\infty}x_n \text{ exists} \}\\ c_0(K)=\{ x\in c(K)\mid \lim_{n\to\infty}x_n=0 \}\\ l_\infty(K)=\{x:\Bbb N\to K\mid x_n \text{ is bounded}\}$$ I want to ...
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23 views

Topologies on Spaces of k-rectifiable sets, and $C^k$ convergence

I've been thinking about topologies on the spaces of k-rectifiable sets (Hausdorff metric topology, varifold topology, etc) in $\mathbb{R}^n$, and I'm wondering if there are hypotheses under which ...
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36 views

Closed continuous operator has closed domain? Question about completeness

This question is about the statement: Let $X$, $Y$ be normed linear spaces and $D$ a linear subspace of $X$ and suppose that $A\colon D \to Y$ is a linear operator. If $A$ is continuous and closed ...
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28 views

Geometric Meaning of Modulus of Smoothness

Could you explain me the "geometric meaning" of the following definition (it's taken from a book on spline functions theory)? Definition: Given $1 \leq p \leq \infty$ a positive integer $r$, and ...
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13 views

closability of $n$-th power of paranormal operator

It it well-known, that there exists closable paranormal operator $A$ such that $\overline{A}$ is not paranormal [1] and if $B$ is paranormal then $B^n$ is also paranormal [2]. Is there any example of ...
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25 views

General derivative of composition

We know that if $G\in \mathcal{C}^1(\mathbb{R})$ such that $G(0)=0$ and if $u \in W^{1,p}(I)$, then $G(u) \in W^{1,p}(I)$ and $(G(u))' = G'(u) u'$. Do we have a similar result for $W^{s,p}(I)$ with ...
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45 views

does there exist a continuous function $f$ such that $l(f) = 4$?

for a continuous function $f$ on $[-\pi,\pi]$ I have a linear functional $l(f) = \int_{-\pi}^\pi (\sin t) f(t) \ dt$ which I have found a bound: $|l(f)| \leq 4 \|f\|_c$ I am asked if it is possible to ...
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30 views

surjectivity in open mapping theorem

The open mapping theorem states that a surjective, bounded, linear operator $T$ between Banach spaces $E,F$ is open. Let $B_k$ denote the open disk with radius $k$ around $0$ and $\overline\cdot$ the ...
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24 views

concomitant and self-adjoint operator

If $Lu = u^{\prime\prime}+\omega^2u$, show that $L$ is formally self-adjoint and the concomitant is $J(u,v)=vu^\prime-uv^\prime$. Moreover, if $u$ is a solution of $Lu=0$ and $v$ is a solution of ...
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19 views

Proof that $c_0$ is separable (with respect to the $l_\infty$ norm)

I'm looking at the proof that $c_0$ is separable, but I don't understand the proof.In the proof below, it first shows that $S$ is separable, where $S$ is: Next, it shows that $S$ is dense in $c_0$. ...
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24 views

On the origin of the notion of polynomial between Banach spaces

I have already asked here a few questions about polynomials in Banach spaces (Counterexample of polynomials in infinite dimensional Banach spaces, Mujica's "Complex analysis in Banach ...
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59 views

Projection operator in Banach space is continuous

Let $(X,||\cdot ||)$ be a Banach space with a Schauder basis, i.e. there exist $e_j \in X$, $j\in \mathbb{N}$, s.t. $||e_j||=1$ for all $j$ and every $x\in X$ can be uniquely represented as ...
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42 views

Proof compactness of adjoint operators

I am trying to understand a proof of the following statement: Given a complex B-space X and a compact operator $T:X\rightarrow X$, the adjoint operator $T^\ast:X^\ast \rightarrow X^\ast$ is compact as ...
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62 views

ODEs explosion times

I was interested in the concept of solutions to 1-D ODEs "exploding", i.e., diverging to $\infty$ in a finite time, and came across this interesting paper on the arXiv: ...
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28 views

Prove inequality to reject differentiability

Suppose there is a function sequence defined on $[0,1]$ and $f_1(t)=t$. For each $f_n(t)$, there is a set of points $T_n=\{0,2^{-n},2\times2^{-n},\cdots,1\}$, such that between each of these points, ...
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48 views

Show that $S_n \to S $(weakly) and $T_n \to T$ strongly implies $S_nT_n \to ST$ weakly

Let $X,Y,Z$ be Banach Spaces. Let $T_n,T \subset BL(X,Y), S_n,S \in BL(Y,Z)$. Show that a) $S_n \to S $(weakly) and $T_n \to T$ (strongly) implies $S_nT_n \to ST$ (weakly) b) $S_n \to S ...
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43 views

A function which determines whether a function is odd, even or none, functional?

I just got introduced to the concept of functionals in Mathematics. From what I understood, a functional is a special kind of function where the domain space is a function space and the range includes ...
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65 views

The reduction of nilpotency order of nilpotent elements of $C^{*}$ algebras

Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$ Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$? Motivation for ...
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25 views

Sobolev space, cut off function

I am looking for a function.  Let $W^{1,2}(\mathbb{R}^{d})$ be $(1,2)$-Sobolev space on $\mathbb{R}^{d}$. Can we construct the following function? $\chi \in W^{1,2}(\mathbb{R}^{d})$ such that ...
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29 views

Prove the uniform convergence of $\sum_{n}^{\infty} \frac {{x}^{n}}{{(1+2x)}^{n}}$ by finding the small tail

Prove the uniform convergence of $\sum_{n}^{\infty} \frac {{x}^{n}}{{(1+2x)}^{n}},x\in[1,\infty )$ Can we prove this by using the definition directly: For every $\epsilon$ there exist N for every ...
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25 views

Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
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33 views

Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
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22 views

prove the continuity of this function

I want to prove that the following function is continuous : $$ V(\phi)=\int_0^1\left[ \frac{a(x)}{2}\phi_x^2(x)-F\left(\phi(x)\right)\right] dx $$ where: $\phi\in H_0^1([0,1)]$ ...
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11 views

Minimizer of transform function

Consider a real-valued function $V(u): \mathbb{R}\rightarrow \mathbb{R}$. Suppose $\exists$ $b:=argmin_u V(u)$. Consider the function $V(\tilde{u})$ with $\tilde{u}=a+\frac{u}{\sqrt{n}}$, $a,u \in ...
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17 views

approximation of weakly differentriable bochner functions

Given a function $u\in L^2(0,T;H^1(\Omega))$ with $u_t\in L^2(0,T;(H^1(\Omega))^*)$. Can we approximate $u$ by functions $u^k$ with $$u^k=\sum\limits_{i=1}^{n(k)}c_i^k\phi_i^k,\text{ where } c_i^k\in ...
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17 views

Provide example for a certain reaarangement invariant Banach space of Lebesque-measurable functions

I look for an example of a rearrangement invariant Banach space X of Lebesque-measurable functions on $(0,1)$, preferably, which meets the following criterion: There exists a function $g>0$ in $X$ ...
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47 views

why dual of $l_1$ norm is $l_\infty$ and vice versa

This might be a very dumb question but I am having a hard time to understand why dual of $l_1$ norm is $l_\infty$ and vice versa. The dual of a norm is denoted $\lVert\cdot\rVert_*$, defined as $$ ...
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32 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
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12 views

$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : the semi group $V(A)$ of equivalent projections (under Murray Von Neumann equivalence) in $M_∞(A)$ is ...
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42 views

Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of ...
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33 views

Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation: $i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields: $i (u_t,v)+\beta ...
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36 views

Definition of essential spectrum?

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My ...
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23 views

How do we prove that a specific kernel is positive definite (case of logarithm)?

I have a problem proving that some specific kernels are positive definite. In general, I can find the answer quickly enough but here I have a specific case involving a logartihm : ...
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18 views

Triangle inequailty for $L^p$ norm to power $p$

I would like to prove the sharp estimate for the $L_p$ norm to power $p$ with $1\leq p <\infty$. What is the constant $C$ here: $$\left\|\sum_{j=1}^Jf_j\right\|^p_p\leq C\sum_{j=1}^J\|f_j\|_p^p$$ ...
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21 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
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27 views

Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$

Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$ Let $\phi ...
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19 views

What it means for a generalized function to be periodic or radially symmetric??

Let $T$ be a generalized function. I need to provide definitions for $T$ to be periodic and radially symmetric. A function (on $\mathbb{R})$is said to be periodic if there exists a $p \in ...
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57 views

Properties of weakly convergent series in Hilbert space

Let $H$ be a Hilbert space and $\{x_n\}_{n=1}^{\infty}$ given sequence of vectors from $H$. Suppose that for every $\{\alpha\}_{n=1}^{\infty}\in \ell^2$ series $\sum_{n=0}^{\infty}\alpha_nx_n$ is ...