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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19 views

Space of function with countable discontinuity.

I am asking this question just to make sure that I am concluding correctly that a function which is zero everywhere except at certain points; is within the set of space of function with countable ...
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0answers
10 views

Is there any variation of Lipschitz continuity, where one can bound difference between value of 2 functions which act on different space?

Lipschitz continuity can be used to bound the difference between value of a function at two different points. Is there any variation of this, where one can bound difference between value of 2 ...
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0answers
37 views

Sofic groups alternative definition

I am trying to solve exercise two from the following document : http://mtm.ufsc.br/~daemi/soficworkshop/Course%20notes/Lupini%20Lecture%202.pdf I suspect there is an error in the exercise, but I'm ...
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0answers
36 views

Continuity of derivative operator

How to prove that the derivative operator is a continuous linear operator on the space of distributions?
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0answers
19 views

A question involving duality maps

Show that if X is an infinit dimensional and smooth Banach space, then there are no compact duality maps on X. Can someone, please, give me a hint on how to deduce this from the following fact: Let ...
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1answer
17 views

Topology induced by seminorms and initial topology

Let's say we have a family of seminorms $(\rho_\alpha)_{\alpha \in A}$ on a vector space $V$. There are two ways to topologize $V$ using those seminorms: We define topology $\mathcal S$ by a ...
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17 views

functional analysis [closed]

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.9 problem 4 let $\{f_1,f_2,f_3\}$ be the dual basis of $\{e_1,e_2,e_3\}$ for $R^3$ where ...
3
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1answer
160 views

A counter example

I have this two spaces $$C_{\theta}=\{u\in C(\overline{\Omega}), \sup (|x|^{\theta} |u(x)|)<\infty\}$$ with the norm $\displaystyle\|u \|_{\theta}=\sup_{\Omega}(|x|^{\theta} |u(x)|)$ and ...
2
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1answer
36 views

Requirements for the principle of uniform boundedness

The version of the principle of uniform boundedness as we stated it in the lecture seems wrong to me in multiple points. Here is how I would state and proof the principle in the terms we used in the ...
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1answer
82 views

Proving that T(t)x is in the domain

$(T(t))_{t\ge 0}$ is a $C_{0}$-semigroup on a Banach space $X$ with generator $A:D(A)\subset X\to X$. For $k\ge 2$, define $$D(A^{k}):=\{x\in D(A^{k-1})|A^{k-1}x\in D(A)\}$$ I want to show that for ...
4
votes
1answer
43 views

A necessary and sufficient criterion for an element of a multiplier $ C^{*} $-algebra to be positive.

I am trying to find a reference for the following assertion: Let $ A $ be a $ C^{*} $-algebra, and let $ M(A) $ denote its multiplier algebra. Then $ m $ is a positive element of $ M(A) $ if and ...
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1answer
47 views

Space of functions on $[a, b]$ with countable discontinuity

We are quite familiar with $C([a,b])$, the set of all continuous functions defined on a closed interval $[a,b]$ with the supremum norm. My question is; if we allow function with countable ...
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1answer
28 views

Essentially self adjoint operator

Given a linear operator a self adjoint operator A
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1answer
31 views

Need help understanding compact embedding of hilbert spaces

I am trying to understand the following statement, and I would like some clarification Consider a Hilbert space $H$ which is compactly embedded in a Hilbert space $L$, with $H^*$ being the dual ...
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1answer
15 views

It is true that the relatively compact open subsets cannot exist in infinite-dimensional normed spaces?

It is true that the relatively compact open subsets cannot exist in infinite-dimensional normed spaces? Why yes / not? Can someone,please, explain to me? Thank you!
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1answer
36 views

monotone convergence theorem( converges in measure)

I have heard that the monotone convergence theorem hold if almost everywhere convergence is replaced by convergence in measure. I concur if fn converge in measure then there exists a subsequence ...
1
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1answer
23 views

Uniform Convergence on Compact Sets Means Uniform Convergence on the whole Set

Let $\Omega\in Open(\mathbb{R}^n)$ for some $n\in\mathbb{N}_{\geq1}$. Then we know that $\exists \{K_n\}_{n\in\mathbb{N}_{\geq1}}$, a collection of compact sets, such that $\Omega = ...
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1answer
36 views

Showing Sobolev space $W^{1,2}$ is a Hilbert space

I have the Sobolev space $W^{1,2}$ consisting of all continuous functions $f \in L^2(\mathbb{R})$ such that there exists an $f'$ with $f(b) - f(a) = \int_a ^b f'(t) dt$. $W^{1,2}$ has inner product ...
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0answers
29 views

Compact operators and functions

Assume we have a look to the space $L^2(S^1)$. An orthonormal basis of $L^2(S^1)$ is given by $p_n(x)=e^{2\pi in x}$ $(n\in\Bbb{Z})$. One can also have a look at the operator $S(p_n)=sgn(n)\cdot p_n$ ...
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0answers
8 views

weak* topology and orthogonal space [duplicate]

Let $E$ be a Banach space;How can you prove that for $M \subset E$ be a linear subspace, and $f_0 \in E^*$ there exists some $g_0\in M^\perp$ such that $$\inf_{g\in M^\perp} ||f_0 − g||=||f_0 − ...
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0answers
23 views

Orthogonal and weak star topology [closed]

Let $E$ be a Banach space; let $M \subset E$ be a linear subspace, and let $f_0 \in E^*$. Prove that there exists some $g_0\in M^\perp$ such that $$\inf_{g\in M^\perp} ||f_0 − g||=||f_0 − g_0||.$$
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1answer
42 views

Extension of bounded operators between norm spaces

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then, this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le ...
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0answers
28 views

Borel functional calculus and multiplication operator

Let $A_f$ be the multiplication operator in $L^2(\mathbb R)$ with the function $f$. If $g$ is a bounded Borel function on $\mathbb R$, why is $g(A_f)$ defined by the functional calculus the ...
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1answer
21 views

Affine, surjective map between convex sets

The setup for my question is the following: I have a compact and convex subset $K$ of some locally convex topological vector space. Within $K$ there is a $T\subset K$ which is compact and convex and ...
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1answer
50 views

Image of collection of probability measures in $C_b(S)'$

Let $(S,d)$ be a Polisch space (i.e. a complete and separable metric space) and $\mathcal{P}$ the collection of probability measures on the borel sigma algebra of $(S,d)$ which we denote by ...
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1answer
23 views

Projection valued measure of bounded self-adjoint operator.

Let $A$ be a bounded self-adjoint operator with $P_E=\chi_E(A)$ as its projection valued measure on set $E\subset \mathbb{R}$, then $f(A)=\int f(\lambda)dP_\lambda$ and $A=\int \lambda dP_\lambda$. ...
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1answer
45 views

Self adjoint operator property

Let $A$ and $B$ be two self adjoint operators on $L^2(\mathbb{R}, \mu)$ and $L^2(\mathbb{R}, \gamma)$, suppose the spectral measure $\mu, \gamma$ are absolutely continuous. Show that $A$ and $B$ are ...
4
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1answer
65 views

Geometrical representation of the unit ball?

Let $E$ be the vector space of $\mathbb{R}$-valued continuous functions on $[0\ 1]$. With the norm $\| f \| = \max \{\ | f (x) |; 0 \leq x \leq 1\}$, the open ball centered at $f$ and radius $r$ has ...
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1answer
54 views

Prove a function has a maximum and minimum along a domain

Given the function $f:[13,132] \to R$ defined by $f(x)=sinx+x^3-$2 $e^x $ prove that the function has a maximum and minimum along the domain. I understand that a function has a maximum and minimum ...
2
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0answers
27 views

positive elements in $L(H)$

At the moment I'm studying positive elements in $C^*$-algebras. Let $H$ be a complex Hilbert space, $L(H)$ the linear bounded operators $H\to H$ and $T^*=T$. Then: $$\sigma(T)\subseteq [0,\|T\| ]\; ...
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21 views

Need simple logic or formula for the below problem!

The problem is simple tip calculator here calculating remaining tip from the money got from user. Inputs - x,y,z Where "x,y" are two denominations of currency and "z" is billamount If x = 2, y=5, ...
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1answer
17 views

Dual space of a finite dimensional

Let $V$ be a normed space with dual $V^*$. Then $V$ is finite dimensional if and only if $V^*$ is finite dimensional, and in fact $\dim{V} =\dim{V^*}$ I set up the proof as follows: since $V$ is ...
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0answers
21 views

Does any continuous map between two Banach spaces over a nonarchimedian field is a closed map?

Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a ...
2
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1answer
24 views

Dual space of a finite dimensional is finite dimensional

Let $V$ be a normed space with dual $V^*$. Then $E$ is finite dimensional if and only if $V^*$ is finite dimensional, and in fact $\dim{V} =\dim{V^*}$. I set up the proof as follows: Let ...
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1answer
46 views

problems in the book A Course in Functional Analysis - John B. Conway [closed]

I meet two problems. Can people solve them clearly, please! Q1: Let $M = \{x \in \ell_p: x(2n)=0 \forall n\}$, where $1\le p \le \infty$. Show that $\ell _p/M$ is isometrically isomorphic to ...
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0answers
25 views

How to prove $f$ is outer, when $Re f$ >0?

This is an exercise problem, which can be found in most of the book which deals with Hardy Space. Any kind of suggestion or hint is also welcome. The question is following: Let $f$ be a function in ...
2
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2answers
101 views

Prove that $T(B)$ is relatively compact in $C([a,b])$.

Let $B$ be the unit ball in $C([a,b])$. Define for $f\in C([a,b])$, $$Tf(x)=\int_a^b (-x^2+e^{-x^2+y})f(y)dy.$$ Prove that $T(B)$ is relatively compact in $C([a,b])$. My attempt: If $|f(x)| \le ...
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0answers
34 views

Show that the functional is continuous everywhere in $V$

Let $J: V \to \mathbb{R}$ be a linear functional and $V$ a linear space with norm. Show that if $J$ is continuous on $0 \in V$ then $J$ is continuous everywhere in $V$. That's what I have tried: ...
2
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2answers
44 views

Intersection of Hilbert spaces

Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ ...
4
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2answers
101 views

Compute Quotient Space

I have been struggling with this computation for a while now. I thought I was almost there, but it now results I still have nothing. So here is the initial problem: Let $c=\left\{ (x_j)_j \subset ...
2
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0answers
34 views

independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper: Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...
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1answer
28 views

continuous functional calculus for nonunital $c^*$-algebras

In lecture we had the continuous functional calculus for unital $c^*$-algebras: Let $A$ be an unital $c^*$-algebra, $a\in A$ normal and let $$alg(a,a^*)=\overline{ \{ ...
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1answer
19 views

Theorem 3.3-1, Lemma 3.3-2, and Theorem 3.3-4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to write these as one?

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: (First, I'm giving ...
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0answers
14 views

Is the set of $m\times n$ matrices whose entries are bounded by $1$ in absolute compact in $C^1(R^n,R^m)$? [closed]

It is well known that each linear map $L:R^n\to R^m$ can be given by an $m\times n$-matrix. Let $C$ be the set of linear transformations with the absolute values of the entries of the matrix not ...
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2answers
39 views

Norm inequality

While trying to compute a quotient space, the next problem has come to my attention: Let $x=(x_j)_j$ and $y=(y_j)_j$ be two complex convergent sequences such that $x-y=(x_j-y_j)_j$, is a constant ...
1
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1answer
24 views

Degree of map on $U(n)$ and roots in $U(n)$

Recently I went to a talk of A.Thom in which he sketched a proof of the fact that the groups U(n) satisfy the Kervaire-Laudenbach conjecture. At some point in the proof you have to argue that the map ...
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1answer
34 views

Eigenvector of a $C^n$ class matrix

Let $A$ be the following matrix function: $\Bbb{R} \to \Bbb{R}^{a \times (a+1)}$ $t \mapsto A(t)$ Let us suppose that $A$ is $C^{\infty}$, meaning that all of $A$ coefficients are $C^{\infty}$. Let ...
2
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5answers
40 views

Equivalent Norms $\|x\|_1=\|x\|+|f(x)|$

Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous. I've ...
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0answers
47 views

functional inequality with a strange constant at RHS

I want to prove the following result. Let consider a function $f$ twice continuously differentiable from $[0,1]$ into $\mathbb{R}$ such that ...
2
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0answers
29 views

Hamiltonian: Invariant Core

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote its evolution by: ...