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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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
38 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 = ...
2
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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
25 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
44 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
29 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
28 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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0answers
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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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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1answer
42 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: ...
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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
102 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
35 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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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 ...
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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 ...
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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
49 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: ...
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1answer
18 views

Reducing Subspaces: Hamiltonian

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a projection: $$P\in\mathcal{B}(\mathcal{H}):\quad P^2=P=P^*$$ Then one has: ...
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1answer
30 views

Quotient space of infinite dimensional vector space

On an exam today I used that if $X=\mathcal{C}[a,b]$ and $Y=\{f\in X : f(a)=f(b)\}$, then the projection $\pi: X\rightarrow X/Y$ has the property $\ker(\pi)=Y$. This led me to the following: Suppose ...
2
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0answers
23 views

weakly open subset in $M[0,1]$ (the space of finite measures on $[0,1]$)

I came across this question when reading Lynch and Sethuraman (1987): Large deviations for processes with independent increments. Let $M[0,1]$ be the space of finite nonnegative measures on $[0,1]$ ...
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0answers
34 views

Compactification of Polish space , What is its use?

I want to know the use of the fact that a Polish space can be homeomorphically embedded into a dense subset of a compact metric space. For example, a continuous function $f$ on a Polish space can't ...
2
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1answer
25 views

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete.

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete. I've already seen that this question has been asked, and already answered, however, I've managed ...
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0answers
33 views

Idea of the proof of Lusternik-Schnirelmann

I have this theorem of Lusternik-Schnirelmann from Chang's book: " Let $M$ be a smooth Banach-Finsler manifold. Suppose that $f\in C^1(M,\mathbb{R})$ is a function bounded from below, satisfying ...
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2answers
31 views

Periodic function

Considering the function $F(x)=x-E(x)$ such as $E(x)$ is the integer part of $x$ So here just with observation we can see that : $f(x+1)=(x+1)-E(x+1)=f(x)$. But here mathematics is not based just ...
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1answer
19 views

Selfadjoint Operators: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Regard a sequence: ...
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1answer
35 views

Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, ...
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0answers
18 views

Integral kernels of self-adjoint operators

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...
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1answer
52 views

A copy of $l_\infty$ in a infinite dimensional Banach space

Let $E$ an infinite dimensional Banach space. Using the Hahn-Banach extension theorem, prove that there is a sequence $(y_n)\subset E$ and a decreasing sequence of closed subspaces ...
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0answers
66 views

Existing complete function space allowing discontinuity .

This is a question which came to me due to several previous question: sorry for the all previous links necessary to look to get the question. The latest question is in the link: Convergence on Norm ...
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0answers
40 views

What's the maximum speed of snake so that the frog can escape?

Suppose there's a round pond, a frog which can swim as 1 meter / second, and a snake that moves along the pond ridge but cannot swim. If the frog can reach any point on the ridge of the pond before ...
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1answer
38 views

conditions for norm of linear bounded operator to satisfy $\lvert T_x (y) \rvert = \lVert T_x \rVert$.

Let $x = (x_n)_{n \in \mathbb N} \in l^\infty$ and let $T_x : l^1 \rightarrow \mathbb F$ be defined by $T_x (y) = \sum_{n=1}^\infty x_ny_n$. What condition on $x$ is needed so that there exists $y \in ...
4
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0answers
73 views

Convergence on Norm vector space.

I am not sure if this question make sense mathematically, so please bear with my ignorance. This is an extension to the question in the link: Is complete metric space is required? It seems in many ...
3
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2answers
35 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...