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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8 views

locally convex topological vector space using semi norms

Given a vector space and a family of semi-norms defined on it, I have to prove that it becomes a locally convex topological vector space. To prove that it becomes a locally convex space I have to ...
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0answers
9 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply separability?

I want know If $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does a strictly convex and weak metrizable unit ...
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0answers
6 views

An exercise about reduced crossed product of a C*-dynamical system

Exercise 4.1.3. Let $A$ and $B$ be two C*-algebras and $\Gamma$ be a discrete group. If $\alpha:\Gamma\rightarrow Aut(A)$ is an action and $\tau\otimes\alpha:\Gamma\rightarrow Aut(B\otimes A)$ is ...
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10 views

A question about cross product of C*-dynamical system

Here is a Theorem (P123) in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Theorem 4.2.4. Let $A$ be a C*-algebra and $\mathbb{Z}$ be the set of integers. ...
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0answers
10 views

general existence theorem of nonlinear parabolic PDE on a unit circle

I wish to study existence/uniqueness of the solutions to a system (possibly coupled) of nonlinear PDE arise from biology on a unit circle. Could any one suggest me any references for studying ...
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0answers
15 views

Prove that seminorm determines topology

To prove seminorm determines topology first open set is defined in terms of semi norms. This is understandable. Then he proves that the set $\{x : p_{\gamma}(x) <c\}$ is open and that for every ...
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0answers
23 views

Hilbert subspaces of $B(\mathbb{R}^n)$

Apart from the one-dimensional subspaces, what are the Hilbert subspaces of $B(\mathbb{R}^n)$? I'm not even sure if such subspaces exist.
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0answers
19 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
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1answer
41 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 ...
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1answer
38 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
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1answer
50 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, ...
3
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1answer
26 views

The isomorphism of the Morrey space $L^{2, n} (\Omega)$ and $L^{\infty} \cap L^2 (\Omega)$

The Morrey space $L^{2, \nu} (\Omega)$ for an open set $\Omega \subset \mathbb{R}^n$ and for $1 \leq \nu \leq n$ is defined as $$L^{2, n} (\Omega) = \{ f \in L^2(\Omega); \hspace{2mm} [f]^2_{L^{2, ...
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1answer
21 views

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
3
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1answer
39 views

Find the values of $f(0)$, $f(4)$, $f(6)$ and $f(18)$

A function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(2)=2$ and $$f(x+1)+f(x-1)=\sqrt{3}f(x) \tag{1}.$$ Find the values of $f(0)$, $f(4)$, $f(6)$ and $f(18)$. My approach: replace $x$ ...
2
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0answers
22 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply locally uniform convexity?

I'm trying to find a proof for this question Let $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does $X$ admits ...
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1answer
20 views

Terminology: 'pointwise monotone functional'?

I have a set $\mathcal{F}$ of real-valued functions, $$f_i(\cdot):\mathbb{R}\to\mathbb{R} \, ,$$ and a (linear) functional $T$ defined on $\mathcal{F}$, $$T:f_i \mapsto T[f_i] \in \mathbb{R} \, ,$$ ...
2
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3answers
49 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
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1answer
38 views

Question on the proof of Open mapping Theorem

I am studying on Open Mapping Theorem. I am stuck at a point in the proof. Open Unit Ball: A bounded linear operator T from a Banach space X onto a Banach space Y has the property that the image ...
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0answers
32 views

Existence of a semigroup of bounded operators which is not $C_0$

Let $X$ be any Banach space. Then we can define a $C_0 $ semi group of bounded operators on $X$. But my question is that can we define a semi group of bounded operators which is not $C_0$?
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0answers
20 views

Continuity of Function Related to F-norms

Let $X$ be a locally bounded $F$-space and $\left\|\cdot\right\|$ be an $F$-norm on $X$. Suppose that $\left\|\cdot\right\|$ is concave: for all $x\in X$ fixed, the function ...
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0answers
11 views

Maximizing the “uniformity” of a distribution subject to moment constraints

I want to develop a continuous probability density, subject to moment constraints, that is maximally "uniform". A maximally uniform density is a density that has the smallest maximum probability ...
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0answers
44 views

Existance of inverse of an operator

I was studying abstract inverse source problem of an abstract heat equation in approach of semi group theory. There I am unable to find the reason of existence of inverse of an operator that i have ...
2
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2answers
30 views

Boundedness of Volterra operator with Sobolev norm

Consider the subspace of $C^\infty([0,1])$ functions in the Sobolev space $H^1$. I want to know whether the Volterra operator \begin{equation} V(f)(t) = \int_0^t f(s) \, ds \end{equation} is bounded ...
4
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2answers
52 views

Riesz's Lemma for $l^\infty$ and $\alpha = 1$

Riesz's Lemma says the following: Let $X$ be a normed vector space and $Y$ a proper closed subspace of $X$. Pick $\alpha \in (0,1)$. Then $\exists x\in X$ such that $|x|=1$ and $d(x,y) \geq \alpha$ ...
3
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1answer
32 views

Is $C^\infty_0$ dense in $C^\infty$ w.r.t. $\|\cdot\|_{L^p}$ and $\|\cdot\|_{W^{1,p}}$?

Is the space $C^\infty_0(\Omega)$ of smooth functions with bounded support, dense in the set of smooth functions $C^\infty(\Omega)$ with respect to the norms $\|\cdot\|_{L^p}$ and the Sobolev-Norm ...
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1answer
28 views

The image of non-reflexive Banach space in its bidual is at large distance from some unit vector in the bidual

I am having some difficulties with part of a problem, I am working on. Let $X$ be a non-reflexive Banach space and let $i: X \to X^{**}$ be the canonical embedding. Show that for given $\epsilon ...
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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. ...
2
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1answer
58 views

Convex interior topology

I have found a fascinating example of topology on a vector space $V$, but I cannot prove its interesting properties to myself. Let $\mathcal{B}$ be the family of all convex symmetric (i.e. $\forall ...
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1answer
61 views

Weak convergence of scaled elements implies norm convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
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1answer
36 views

The symetric operator is linear operator. [on hold]

Definition: Let be $X$ unitary space. Operator $A:X\rightarrow X$ called symetric if $$ (Ax\vert y)=(x\vert Ay), (x,y\in X).$$ I saw some books functional analysis but can not find verification of ...
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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 ...
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1answer
40 views

Difference between Half Quadratic vs Quadratic

Half quadratic minimization/penalty/optimization, I am unable to find any related material/resources. If anyone can point to some useful resources, it will be great
6
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3answers
222 views

Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
2
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2answers
40 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. ...
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1answer
42 views

Characterizing C* algebra generated by elements.

Let $A$ be a C*-algebra and $A_0\subset A$. Then it is known that the $\mathbb{C}$-algebra generated by $A_0$ (i.e. the intersection of all sub-$\mathbb{C}$-algebras containing it ) is just the ...
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1answer
52 views

Explaining why $\Vert x_n\Vert _1=\frac{3}{4}$

Let $X_1$ respectively $X_2$ denote the space $C [a, b]$, $a <b$ with norm: $$\Vert x\Vert_1=\int_a^b \vert x(t)\vert dt; $$$$\Vert x\Vert_2=\left(\int_a^b \vert x(t)^2\vert ...
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0answers
37 views

Formal definition of “node” with respect to eigenvalues and functional analysis.

I'm concerned with a special problem of spectral analysis for a certain Sturm-Liouville-differential-operator, that is to say $L:=\frac{d^2}{dx^2}-q(x)$ and the spectrum $\sigma(L)$. While reading an ...
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0answers
39 views

Calcul of limit [on hold]

What is the limit of $$\lim_{f \rightarrow 0} \frac{ \nabla {f(x)} }{\sin{(f(x))}}?$$ We can use the Poincare inequality and the famous limits: $$\lim_{x\rightarrow 0} ...
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0answers
17 views

Convex Inequality describing Functions inside specific area

Let us assume that we have two functions $f_1$, $f_2:[0,1] \rightarrow \mathbb{R}^{2}$, which describe each a point trajectory on the plane. Let us further assume that we parametrize those functions ...
2
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0answers
25 views

Resolvent and spectrum of a self-adjoint extension

In this paper, they give the resolvent, spectrum, and eigenfunctions of the self-adjoint extension of the Laplacian on a rectangle that corresponds to a delta potential at an arbitrary point (items ...
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0answers
22 views

Get locally uniformly convex norm by bounded linear operator

I want to prove this theorem Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for every bounded ...
4
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2answers
66 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...
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1answer
49 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 ...
2
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0answers
41 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
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1answer
43 views

Orthogonal basis of complete Euclidean space

friends! I read that any complete Euclidean, complex or real, space $R$ has a (normalized) orthogonal basis. By orthogonal basis an orthogonal system of vectors such that the smallest closed subspace ...
2
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1answer
24 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
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0answers
34 views

Book suggestion functional analysis [duplicate]

I am studding functional analysis and applications. Does anyone have a good recommendation of books//lectures/resources/etc.? Thanks.
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0answers
26 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
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2answers
56 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
2
votes
1answer
26 views

Kernel of the Extension of a Bounded Linear Operator

Suppose $T\colon E\to F$ is a bounded linear operator between Banach spaces. Moreover let $i\colon E\to E’$ be a dense, compact inclusion of $E$ into some other Banach space $E’$. Finally assume that ...