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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66 views

Is adjoint operator of generator of an analytic semigroup be a generator of analytic semi-group?

Let $X$ be a Banach space. The adjoint semigroup $\{T(t)^\prime:t\ge 0\}$ consisting of all adjoint operators $T(t)^\prime$ on the dual space $X^\prime$ is, in general, not strongly continuous where ...
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1answer
36 views

what are differences between metric space and metric linear space?

A metric linear space is a linear space equipped with metric but i want to know the point wise differences between metric space and metric linear space.Can any body write it down in points?
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0answers
67 views

Herz-Schur multiplier bounded if corresponding functional is bounded

I want to prove the following statement: Let $\Gamma$ be a discrete group and $\phi:\Gamma\rightarrow\mathbb{C}$ a function and ...
12
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4answers
1k views

Why is an infinite dimensional space so different than a finite dimensional one?

In functional analysis there is a big difference between finite- and infinite-dimensional vector spaces. I have found other questions with nice answers here and here. However, I don't grasp the ...
5
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1answer
146 views

Comparison between weak convergences in Banach spaces

Let $X$ be a Banach space and let $Y=BC(\mathbb{R},X)$ be the Banach space of all bounded continuous functions from $\mathbb{R}$ to $X$ equipped with the supremum norm. Let $(f_n)_n$ be a sequence of ...
3
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0answers
111 views

Small question about condition in the Bouchala paper's 2005

I have this condition from this paper: http://ejde.math.txstate.edu/Volumes/2005/08/bouchala.pdf (Strong resonance problems for the one-dimensional $p$-Laplacian. Bouchala, Jiri. Electronic Journal ...
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1answer
26 views

Dimension of quotient normed linear space

Suppose $M$ is a normed linear space. $L$ and $N$ are two closed subspaces of $M$ such that $L \subseteq N$. Then $L$ is a closed subspace of $N$. Let $\text{dim}(M/N)=r$ and $\text{dim}(N/L)=s$. My ...
2
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2answers
933 views

Every bounded sequence has a weakly convergent subsequence: salvage this proof?

I tried to prove the following theorem and was wondering if someone could please tell me if my proof can be fixed somehow... Theorem: Let $H$ be a Hilbert space and $x_n\in H$ a bounded sequence. ...
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1answer
58 views

Closure of the interior of the epigraph

Suppose $f:E\to(-\infty,\infty]$, where $E$ is a Banach space, is lower semi-continuous, convex, and the interior of $epi(f)\neq\emptyset$. Show that $\overline{int(epi(f))} = epi(f)$ \begin{equation} ...
2
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1answer
165 views

On the Definition of Gateaux Derivative

My question is about two different definitions Gateaux derivative. I have seen the following two definitions but whether they are equivalent or which one is better to use I am not sure about: ...
0
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1answer
110 views

Spectral norm of submatrices of a matrix with bounded spectral norm and maximum-entry

Let $A(t)=A$ be a symmetric, positive-definite matrix in $\mathbb{R}^{p\times p}.$ Suppose that the maximum-magnitude entry of $A_t$ has magnitude bounded above by $f(t)$. Suppose also that the ...
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1answer
52 views

a question about Riesz's theorem

Riesz's theorem: Let $(V,\|\cdot\|)$ be a normed vector space, and suppose $C$ is a compact subset of $V$, moreover, $C$'s interior is not empty, and then please prove $\dim(V)<\infty$. ...
2
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1answer
58 views

Show that $X$ is Banach space and describe $X^*$.

Let $X=L^2(\mu)\times L^2(\mu)=\{(f,g)|f,g\in L^2(\mu)\}$ be the linear space normed by $\|(f,g)\|=(\|f\|_2^3+\|g\|_2^3)^{1/3}$. Show that $X$ is Banach space and describe $X^*$. My Work: We ...
2
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0answers
16 views

relating to $L_{p}$-norm [duplicate]

Let $(X, \Sigma, \mu)$ be a measure space and $f$ be a measurable function. For each $1\le p<+\infty$, setting $$\|f\|_{p}:=\left(\int\limits_{X}|f|^{p}\, \mathrm{d}\mu\right)^{\frac{1}{p}} ...
1
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0answers
32 views

Step function scalar product inequality

I would like to prove the following inequality: $$\langle f,\frac{|N.+1|}{N} \rangle^2 \leq \langle f,. \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i ...
1
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1answer
23 views

Question about the following proof in Hilbert space

I started reading the book "Mixed Finite Element Technologies" by Peter Wriggers and Carsten Carstensen, and I have a question about the following. Here is the setup: Then, the authors prove the ...
4
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2answers
138 views

Do $L^p$ spaces have the approximation property?

A Banach space $X$ has the approximation property if every compact operator $T:X \to X$ is the norm-limit of a sequence of finite-rank operators. My question is if there is a simple proof that the ...
0
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1answer
75 views

Weak topology coarser than norm topology

In these lecture notes on page 71(Example 5.29 a) it is claimed that for a LCS X the weak topology is coarser than the topology of the LCS, but afaik this does not even hold for normed spaces or am I ...
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1answer
42 views

Is a cyclic subspace of a compact unitary representation finite dimensional?

Let $K$ be a compact Lie group and let $\rho_k: H \rightarrow H$ be a (strongly continuous) unitary representation of K on a Hilbert space H. Why does the orbit, $\rho(K)v$ ,of any $v\in H$ generate a ...
0
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1answer
41 views

Feeling behind Krein-Milman theorem

We said that the Krein Milman theorem is valid in a LCS $X$ for non-empty convex and compact sets $K$ and it tells us that: 1.) ex(K) $\neq \emptyset$ 2.) $K = \overline{co}(ex(K))$. 3.) If $K = ...
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1answer
65 views

Can continuous functions with range in the unit circle be the average of two different continuous functions with the same range? [duplicate]

Suppose $f:[0,1]\to {\Bbb C}$ is a continous function such that $|f(x)|=1$ for all $x\in[0,1]$. Does there exist different continuous functions $f_1,f_2:[0,1]\to{\Bbb C}$ such that $f=(f_1+f_2)/2$ ...
2
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1answer
54 views

Mourre Theory: Resolvent Formula

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its resolvent by: $$z\in\rho(H):\quad R(z):=(z-H)^{-1}$$ Introduce its ...
6
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1answer
73 views

Open neighbourhoods in topological vector spaces

It is well known that each open ball in a Banach space is homeomorphic to the whole space. Can we extend this to topological vector spaces? In other words, does every non-void open set in a ...
2
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1answer
41 views

Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$

Let $X$ be a normed linear space and $M$ be a proper closed linear subspace of $X$. Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$ My Work: Let $ ϵ>0$. Since ...
1
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2answers
86 views

Definition of operator norm

I want to show $T=d/dx$ is unbounded on $C^1[a,b]$ with $b>1$. Take a sequence $f(x)=x^n$, and $\|T\|=\sup_{x\in[a,b]}\frac{\|Tx\|}{\|x\|}=\frac{\|n\cdot b^{n-1}\|}{\|b\|}$. I want to claim as $n$ ...
2
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0answers
43 views

Exercise in Hahn-Banach Theorem; Finding linear functional $-p(-x)\leq f(x)\leq p(x)$

(The following exercises are in Kreyszig's book 218 page; EXE 10) I want to solve the following exercise : If $X=l^\infty$, let $p(x)=\lim\sup x_i $, whichi is sublinear. Then find a linear functional ...
1
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2answers
20 views

Why $\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$?

Let $f,g\in L^2$ with Lebesgue measure. and $K:L^2\to L^2$ be some linear and continuous operator. Show that $$\|f-g\| \leq \sup_{h\in H}\frac{\|h\|}{\|Kh\|}\|K(f-g)\|$$ where $h\in H\subset L^2$.
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1answer
46 views

Question about bilinear form on Hilbert space

I am trying to verify the following Let $H$ be a Hilbert space, and let $a(\cdot,\cdot)$ be a real continuous bilinear form on $H$ Then, define the operator $A:H-> H'$ as $Au(v) :=a(u,v), v\in H$ ...
3
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2answers
45 views

If $T^2=T$ then determine whether $\ker T=\operatorname{Range}\,(T)^\perp$.

Let $T$ be linear operator on a finite dimensional inner product space $V$ such that $T^2=T$. Determine whether $\ker T=\operatorname{Range}\,(T)^\perp$. I have proved that $\ker ...
0
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0answers
47 views

Finding general orthogonal polynomials

Many Special functions are orthogonal; for example, the sine and cosine function is an orthogonal function. Also, a couple of orthogonal polynomials are well-known. Now I'm asking the following: Given ...
0
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2answers
69 views

Question about dual of dual of Hilbert space

Let $H$ be a real Hilbert space and let $H'$ be the set of continuous linear functionals on $H$. Then, I know by the Riesz Theorem that for every $L(\cdot) \in H'$, there exists a unique $u\in H$ so ...
0
votes
2answers
291 views

Show that if the product (the composite) of two linear operators exist, it is linear

I am given this problem to consider and I am unsure how to prove it. It would be nice to see a definition for what it means to be linear or how to check if something is linear because I don't know ...
3
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1answer
108 views

About a technique used in the proof of Hahn-Banach Theorem

Recall Hahn-Banach (cf. Kreyszig's book) : If $X$ is a real vector space with a sublinear functional $p$ and if $f$ is linear on a subspace $Z$ with $p(z)\geq f(z),\ z\in Z$, then there exists an ...
2
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1answer
84 views

Function decomposition in harmonic analysis

In the book "Weighted Norm Inequalities and Related Topics" by José García-Cuerva, J.-L. Rubio de Francia page 144 it was shown that for a measurable function $f$ and $t>0$ $$ |E_t|\leq ...
0
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0answers
39 views

Convergence in Bochner space (don't follow an argument, can you explain it to me please)

Define $V:=W^{\beta, 2} \subset H:=L^2$ which is compact and dense. It follows that $L^2(0,T;V) \subset L^2(0,T;H)$. Let $w^\epsilon$ be a sequence which is uniformly bounded in $L^2(0,T;V)$ and in ...
2
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1answer
85 views

Sobolev embedding theorem in the homogeneous case

We know that if $s>\frac{n}{2}$ the following inclusion holds $$H^s(\mathbb{R}^n)\subset L^\infty(\mathbb{R}^n)$$ Is it also true in the case we deal with the homogeneous space ...
1
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1answer
52 views

How do I prove the interior of subspace $\ell^1$?

Let $E:=\ell^1$ is Banach space with standard norm for $\ell^1$, $P:=\{\bar{x}\in\ell^1: \bar{x}=(x_i)=(x_1,x_2,\ldots),x_i \geq 0, \forall i \in \mathbb{N}\}$ and defined that interior of $P$ is ...
0
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1answer
49 views

Find a functional F such that F(g)=<g,h> doesn't hold

Let $H=C[0,1]$ with inner product $<f,g>=\int_0^1f(x)g(x)dx$. Find a functional $F\in H^*$ such that there isn't $h\in H$ satisfying $F(g)=<g,h>$, then justify. My question is is ...
2
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0answers
28 views

Weak convergence and norm convergnce along a subsequnece in $H^1(\Omega)$

Let $\Omega$ be an open subset of $\mathbb{R}^d$. Let $(f_n)_n$ be a sequence in $H^2(\Omega)$. Let $f\in H^2(\Omega)$. Assume that $f_n\rightarrow f$ weakly in $H^1(\Omega)$ and that ...
2
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2answers
46 views

prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$

Let $X$ be a linear normed space over $\mathbb{C}$. If a linear functional $L$ on $X$ is not continuous, prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$ Clearly $\{Lx:\|x\|\leq 1 \}\subseteq ...
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0answers
26 views

On the farthest point of sets

It is known that a subset $K$ of a Hilbert space $H$ is called proximinal if every $x\in H$ has a vector $u_x$ of minimum norm in $K$. i.e $\exists u_x\in K: \|x-u_x\|=\min\limits_{u\in K}\|x-u\|.$ ...
2
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1answer
34 views

Expected value is affine in distribution

How to show that $E_{P_X}[X]$ is affine in $P_X$. That is for two distributions $P_{X,1}$ and $P_{X,2}$ we have that \begin{align*} E_{\alpha P_{X,1}+(1-\alpha) P_{X,2}}[X]=\alpha E_{ ...
1
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1answer
23 views

Show that for every real-valued $L^2$ function $u$ on $S^1$ there is a real-valued $v$ in the same space such that $u + iv\in \widetilde{\mathbf H}^2$

For a homework exercise ($1.8$ in the book An Introduction to Operators on the Hardy-Hilbert Space) I am asked to show Let $u$ be a real-valued function in $L^2(S^1)$. Show that there exists a ...
0
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1answer
51 views

Adjoint operators in Hilbert space

Consider the linear and bounded operators $X$ and $Y$on a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot,\cdot \rangle$. How can I show that $$ \langle XY \boldsymbol{v}, ...
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0answers
49 views

Strichartz estimates for wave equations

Let's consider the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$. Strichartz estimates tell us that $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...
2
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0answers
95 views

Question about Stone-Weierstrass theorem

I have a question about Stone - Weierstrass theorem. In the space $C[0,2\pi]$ of continuous functions on $[0,2\pi]$ with the sup norm. Consider the spaces $M$ of all trigonometric polynomials. It's ...
2
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0answers
31 views

Solve Intergal Equation of form g.u1=Int(K.u2) for u1 and u2

I'm trying to find a solution to a differential equation of an unusual form: $$g(x) u_1(x)=\int_a^b K(x,y) u_2(y) dy$$ where $g(x)$ and $K(x,y)$ are known and $u_1(x)$ and $u_2(x)$ are complex ...
0
votes
0answers
36 views

Is sub-level set of a continuos functional closed?

Let $F[f]$ be a continuos functional on set of continuos functions defined on the interval $[a,b]$. With the norm defined $|f|=\max_{ a \le x \le b} |f(x)|$. How to show that the following sub-level ...
6
votes
3answers
95 views

Why is $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$?

I have to show the identity I wrote in the title: it should be $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$ but some computation ...
1
vote
0answers
30 views

Proving a linear function is bounded using the Baire category theorem (or its consequences)

This is a problem from Folland. Let $\mathcal{X}, \mathcal{Y}$ be Banach spaces. If $T : \mathcal{X} \rightarrow \mathcal{Y}$ is linear and $f \circ T \in \mathcal{X}^*$ for all $f \in ...