Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

1
vote
2answers
20 views

Want to show that sequences in little l 1 can be represented by orthonormal basis

I want to show that for a sequence $a=\{a_i\}^{\infty}_1 \in l^1$ I can write $$a= \sum ^\infty _{i=1} a_i e_i $$ For $\{e_i\}^\infty_1 \in l^1$ where the ith term is 1 and all else zero. I know for ...
1
vote
0answers
9 views

Functional differentiation with Grassmann variables

I want to calculate: $$\frac{\partial}{\partial (x^\rho+i\eta\, \psi^\rho)}[f_\mu(x)+i\eta\, \psi^\nu\partial_\nu f_\mu(x)]$$ where $x^\mu(\tau)$, $\psi^\mu(\tau)$ are commuting and anti-commuting ...
2
votes
1answer
18 views

Hessian of inner product

Let $f:\mathbb R^n \rightarrow \mathbb R^n$ be a $C^2$ functions and consider the function $h:\mathbb R^n \rightarrow \mathbb R$ given by $$h(x):=\langle f(x),f(x) \rangle.$$ I am wondering whether ...
1
vote
0answers
33 views

If $\mathcal D(\overline A)$ is dense, are we able to conclude that $\mathcal D(A)$ is dense?

Let $(\mathcal D(A),A)$ be a closable linear operator on a $\mathbb R$-Banach space $E$ and $(\mathcal D(\overline A),\overline A)$ denote its closure. If $\mathcal D(\overline A)$ is dense, are we ...
1
vote
1answer
46 views

Definition of closed

In topology we say that a set is $A$ is closed if its complement $A^\complement$ is open. Now if we have a set $A$ and an operation $*$ we say that $A$ is closed under this operation if $x,y \in A$ ...
1
vote
1answer
39 views

$H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$?

In a paper I see that the authors used $H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary. I think that this imbedding holds ...
2
votes
1answer
21 views

How are the two versions of the Lumer-Phillips theorem given in the books of Engel/Nagel and Pazy related?

Let $E$ be a $\mathbb R$-Banach space and $(\mathcal D(A),A)$ be a densely-defined dissipative linear operator on $E$. In the book of Engel and Nagel, I've found the following verison of the Lumer-...
3
votes
2answers
78 views

Continuity of Linear Operator Between Hilbert Spaces

Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Let $\mathcal{H}$ be a ...
0
votes
0answers
17 views

Set $X$ such that $2^c \leq |X|\leq |(\ell^\infty)^*|$ [duplicate]

I want to prove that $2^c\leq|(\ell^\infty)^*|$. My question is: Is there a set $X$ with $2^c\leq|X|$ such that $f:X\to(\ell^\infty)^*$ is an injective function or $g:(\ell^\infty)^*\to X$ is a ...
4
votes
1answer
56 views

Suppose $U_1,\dots,U_k$ and $V_1,\dots,V_k$ are $n\times n$ unitary matrices. Show that $\|U_1\cdots U_k-V_1\cdots V_k\|\leq\sum_{i=1}^k\|U_i-V_i\|$

Let $V,W$ be complex inner product spaces. Suppose $T: V \to W$ is a linear map, then we define $$\|T\|:=\sup\{\|Tv\|_{W}:\|v\|_{V}=1\}$$ where $\|v\_{V}\|:=\sqrt{\langle v,v\rangle}$ and $\|Tv\|_{W}...
0
votes
1answer
34 views

Show that there is a solution of the Laplace equation $(\mu-A)p=f$

Let $C_0(\mathbb R)$ denote the space of continuous functions vanishing at infinity equipped with the supremum norm, $B$ be a contractive linear operator on $C_0(\mathbb R)$ and $$Af:=\lambda(Bf-f)\;\;...
4
votes
1answer
40 views

Finite rank operators on Hilbert spaces

Let $H$ be a Hilbert space. Question 1: Are all rank one operators from $H$ to $H$ is of the form $$T:H\rightarrow H, x \mapsto \langle x,u\rangle v $$ For some $u,v \in H$. Question 2:...
0
votes
1answer
31 views

Rudin's functional analysis, theorem 4.23 (existence of a certain sequence)

If $X$ is a Banach space, $T \in \mathcal{B}(X)$, $T$ is compact, and $\lambda \neq 0$, then $T - \lambda I$ has closed range. Proof with questions below Proof: By (d) of Theorem 4.18, $\text{dim }...
1
vote
2answers
17 views

Does positive part preserve Holder continuity?

Let $u^+$ denote the positive part of the function $u$ on a bounded domain $\Omega.$ If $u \in C^{0,\alpha}(\bar \Omega)$, is also $u^+ \in C^{0,\alpha}(\bar \Omega)$?
2
votes
2answers
40 views

Is the norm $||x||:= \sum\limits_{i=1}^{\infty} 2^{-i}|x_i|$ equivalent to usual norm in $l_2$?

Is the norm $$||x||:= \sum_{i=1}^{\infty} 2^{-i}|x_i|$$ equivalent to usual norm in $l_2$? I have already shown that it is a norm, I expect that it is equivalent, but I can not prove it.
1
vote
1answer
25 views

Distance to set

Let $S$ be a non empty set in an inner product space $E$ Show that if $x\in E,z\in E$ and $Re<x-z, y-z> \le0$ for each $y\in S$ then $d(x,S)=||x-z||$ I would like a clue on how to approach ...
0
votes
0answers
16 views

structure of subrepresentations of (infinite) sums of irr. representations

Let $G$ be a (locally compact) group and $ ( \pi_1 , V_{\pi_1} ) , ( \pi_2 , V_{\pi_2} ) , \ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum ...
1
vote
1answer
24 views

How do I interpret the boundedness in this space?

As I was reading the article Non linear elliptic and parabolic equations involving measure data, I came across with the following: the sequence $\{f_n \}$ is bounded in the space $L^1(0,T;W^{-1,...
0
votes
0answers
48 views

A question on linear integral equation about non degenerated bilinear form

Let $X$ be a Banach Space , $X\subseteq H,\bar{X}=H$,where $H$ is a Hilbert space $i:=X\to H$ defined by $i(x)=x$ and is continuous. Define $\langle x,y\rangle=\langle ix,iy\rangle$ then $\langle X,X,...
1
vote
2answers
36 views

Weak continuity of the addition and scalar multiplication

Let $X$ be an infinite dimensional normed vector space. Show that vector addition and scalar multiplication are weakly continuous. $$+:X×X \rightarrow X; +(x,y)=x+y$$ $$•:\mathbb {R}×X \rightarrow X; •...
1
vote
1answer
31 views

Uniform Boundedness of Sequence

Let $c_{0}=\{(x_{i})_{i\geq 1} | x_{i}\in\mathbb{K}\text{ and } \lim_{i\rightarrow\infty}x_{i}=0\}$. Suppose $y_{i}\in\mathbb{K}$, $i=1,2,\ldots$, such that \begin{align} \sum_{i=1}^{\infty}x_{i}y_{i},...
0
votes
1answer
19 views

Equivalent definitions of extreme point

We know that a point $x$ of a convex set $K$ in a normed linear space $X$ is said to be an extreme point if for any $y,z\in K$ and for any $0<\lambda<1$, $$x=\lambda y+(1-\lambda)z,$$ then $x=y=...
1
vote
0answers
13 views

Eigenvalue of algebraic multiplicity $m$ is a pole of the resolvent of order $m$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X)$ be a bounded linear operator. Suppose that for some isolated point $\lambda \in \sigma(T)$ and some $m \in \mathbb{N}$ we have $\ker(T-\lambda I)^m ...
-1
votes
1answer
16 views

Why does the determinant vanishing imply nontrivial solutions for a set of differential equations?

Why does this set of equations have to be singular? If the determinant doesn't vanish, what does this imply? I'm a mathmatically inclined materials science student, so I'll probably understand ...
1
vote
0answers
33 views

Showing $f(x)$ $=$ $\sqrt{x}\sin(\frac{1}{x})$ satisfies a Holder Condition of $\alpha < 1$

I'm learning about functions that satisfy Holder's Condition of order $\alpha$. Specifically, A function $f$ is said to satisfy a Hölder condition of order $\alpha > 0$ if there exist $M$ such ...
0
votes
0answers
23 views

Show ${na_n},{nb_n}$ are bounded [on hold]

Let $\frac{a_0}{2}+\sum_{n=1}^\infty(a_ncos(nx)+b_nsin(nx)$ be the fourier series of a function $f\in BV[-\pi,\pi]$. Show that ${na_n},{nb_n}$ are bounded sequences. I am not sure how to prove this....
3
votes
3answers
82 views

A monotonic function that intersect with all lines in $\mathbb R^2$

Let $f:\mathbb R\to\mathbb R$ be a monotone function. Let $\gamma=\{(x,y)\ |\ y=f(x)\}$ is a curve in $\mathbb R^2$. Does there exists a $f$ such that $\gamma\cap L\neq \emptyset \ \forall L\...
1
vote
0answers
33 views

Why do we need projection in the definition of the Stokes operator?

$\DeclareMathOperator{\div}{div}$ $\def\bu{\mathbf{u}}$ Let $D$ be the square $[0,1]^2$ and consider the following space: $$ V:=\{\bu: \bu\in H^2(D)^2, \div \bu=0, u|_{\partial D}=0 \}. $$ Introduce ...
0
votes
0answers
35 views

Is it possible for a sequence of matrices to have pointwise but not uniform convergence?

Is it possible for a sequence of matrices to have pointwise but no uniform convergence? The norm for the matrices is the operator norm.
0
votes
0answers
28 views

Show that Lipchitz functions space is Banach

Let X be a Banach space. Show that $L = \{ f:X \to \mathbb{R}: f \text{ is lipschitz and } f(0)=0 \}$ with norm: $$||f||_{lip}= sup \left\{ \frac{|f(x)-f(y)|}{||x-y||}; x \neq y \in X \right\}$$ ...
0
votes
1answer
22 views

Determining if $f \in BV([0,1])$.

I am reading about Riemann-Stieltjes Integration in Carother's Real Analysis. We find that functions of Bounded Variation provide us with a rich class of integrators. Therefore, I am trying to learn ...
0
votes
1answer
26 views

Weak convergence to 0 iff bounded and pointwise convergence to 0

I am working on a problem from functional analysis that has me stumped. Let $B$ be a reflexive Banach space on some subset of $\mathbb{R}^n$ s.t. point evaluations are continuous. Show that if $f_n\...
2
votes
1answer
48 views

Show that space is not metrizable

Let $X=C([0,1], \mathbb{R})$, $f \in X$, $\epsilon >0$ and $x_1,...,x_n$ in [0,1]. Consider: $V_{f,x_1,...,x_n,\epsilon}= \{ g \in X : |f(x_i)-g(x_i)| < \epsilon, i=1,...,n \}$ and $\tau= \{ ...
2
votes
0answers
33 views

Every Banach limit on $l_{\mathbb{C}}^{\infty}(\mathbb{N})$ is an extension of some Banach limit on $l_{\mathbb{R}}^{\infty}(\mathbb{N})$

Let $l_{\mathbb{C}}^{\infty}(\mathbb{N})$ be the space of bounded complex-valued sequences, $l_{\mathbb{R}}^{\infty}(\mathbb{N})$ the subspace of real-valued sequences. Given any Banach limit $L_1: l_{...
2
votes
1answer
45 views

Convergent sequence of functions and compact set contains the spectrum of the limit

Let $K\subseteq \mathbb{C}$ be a compact subset. Let $(x_n)$ be a convergent sequence of normal elements in a unital $C^*$-algebra $\mathcal{A}$, with limit $x$, such that $\sigma(x_n)\subseteq K$ ...
1
vote
0answers
32 views
+50

If $\kappa$ is a contractive operator on $C_0(ℝ)$, is $λ(κ-\text{id})$ the generator of a Feller semigroup?

Let $E$ be a locally Hausdorff space, $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\},$$ $\kappa$ be a Markov kernel on $(E,\mathcal B(E))...
0
votes
0answers
51 views

Is the integral equal to zero?

Consider $$\int_{S_r} \left(\frac{\partial\phi}{\partial\eta}\right)^2 (x\cdot\eta)\ d\sigma $$ Where $S_r$ is the sphere of radius $r$ and centered at zero, $\phi\in L^2(\mathbb{R}^n)$ and $\eta$ is ...
1
vote
2answers
35 views

Sufficiency in the proof that $L^p(\mu)$ is complete

In the proof that $L^p(\mu)$ is complete for $p\in[1,\infty]$ (as done in Saxe, Theorem 3.21 or in Folland, Theorem 6.6, the latter of which is outlined here) we make use of the following completeness ...
0
votes
1answer
55 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
1
vote
1answer
60 views

Hilbert Transform: limit of xHf(x)

In Terence Tao's notes page 1, cited below, he mentions that it is easy to see that $\lim_{|x| \to \infty} xHf(x) = \frac{1}{\pi}\int f$ where $f$ is a Schwartz function and $H$ is the Hilbert ...
0
votes
0answers
13 views

What is the defining properties of the heat kernel?

According to https://en.wikipedia.org/wiki/Fundamental_solution, the fundamental solution $u$ is defined as the solution of $Pu = \delta$. In the meantime, we know that the fundamental solution of the ...
1
vote
1answer
52 views

Riemann integrability and discontinuity

$g:[0,1] \rightarrow \mathbb{R}$ bounded and $\alpha:[0,1] \rightarrow \mathbb{R}$ non-decreasing. Assume $ g \in \mathbb{R}_\alpha[\delta,1]$ for every $\delta > 0$. I showed that $g\in\mathbb{...
1
vote
1answer
23 views

Convergence of finite dimensional projection of trace class in trace norm

Assume $\mathbb{H}$ is a Hilbert space and $K$ is a trace-class operator on it. Given a fixed ONB $\{e_i\}$ and assume $$K=\sum_{i,j}c_{ij}e_i\otimes e_j.$$ Now, let $K_n = \sum_{1\leq i,j\leq n}c_{...
0
votes
0answers
13 views

How are vector norms and function norms related? [closed]

I'm a little bit confused, so can somebody explain how are vector norms and function norms related?
3
votes
1answer
59 views

An optimization problem in $L^1(0,1)$

Is there any non-negative function $f(t)$ that minimizes $\int_0^1e^{\int_0^tf(s)ds}dt$ and satisfies $\int_0^1sf(s)ds =1$? I guess there is not, because the exponential is minimized if $\int_0^tf(s)...
0
votes
0answers
9 views

The inclusion of Sobolev spaces is compact

I know that the inclusion of Sobolev spaces with compact support is a compact map. Now I wonder whether the inclusion of isotropic Sobolev spaces is compact. My definition of the isotropic Sobolev ...
4
votes
0answers
93 views

If $f$ is integrable, then $\| f\|$ is also integrable.

As usual, a partition of a compact interval $[a, b]$ is, by definition, an strictly increasing family $\Pi = (t_k)_{k = 0}^m$ ($m \geq 0$) of points in the interval such that $t_0 = a$ and $t_m = b;$ $...
1
vote
0answers
80 views
+50

Nonexistance result of elliptic equation

I want to prove that there is no $L^2(\mathbb{R}^N)$-solution of the equation $−\Delta \phi =\lambda \phi$ for every $\lambda \in \mathbb{R}$. I know that the Pohozaev identity asserts for $N\geq 3$, $...
0
votes
1answer
18 views

Closure of Continuously DifferentiableFunctions in Holder Space

Here is a curious (already submitted) homework problem I had in analysis some time ago: Let $\Omega$ be a convex domain in $\mathbb{R}^n$ with $C^1$ boundary. Let $C^{0,\alpha}(\overline{\Omega})$ ...
2
votes
0answers
19 views

If $A \in \mathcal{L}_c(X)$ and $X$ is Banach, then $\dim \ker (\text{id}-A) < + \infty$.

Exercise : Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$. Show that $\dim \ker ( \text{id} - A) < + \infty$. Attempt/Thoughts : The kernel of the operator $(\text{id}-A) : X \to X$ ...