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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9 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
1
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1answer
22 views

Constructing A Space Filling Curve that fills the Unit Square

I'm reading Neal Carothers' Real Analysis, and he constructs a curve defined over $[0,1]$ that fills the unit square as follows: Let $f$ be a real-valued function over $[0,1]$ such that $f$ is $0$ ...
1
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1answer
15 views

Is any bounded linear operator of dual spaces is dual of a linear operator?

Let $X,Y$ be two Banach spaces and $S:Y^* \to X^*$ be a bounded linear operator. Is there always bounded linear $T: X\to Y$ such that $S=T^*?$
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7 views

Suppose $f(x,y)=c(x^{m}+y^{n})-dx^{a}y^{b}+sin(x+y^{2})$, show that f is greater than

Suppose $f(x,y)=c(x^{m}+y^{n})-dx^{a}y^{b}+sin(x+y^{2})$, where m,n are positive even integers, a,b are positive integers, c,d are positive real numbers and $\frac{a}{m}+\frac{b}{n}<1$. Show that ...
1
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0answers
16 views

PDEs, infinite vector spaces, and functional analysis

Suppose I have a PDE for some quantity $q_{t}$:$= q(\boldsymbol{x},t)$ at some time $t$ on a 2D vector space $\boldsymbol{x} = (x_{1},x_{2})$, that looks, for the sake of the question, something like ...
2
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2answers
31 views

Is the composition function again in $L^2[a,b]$ [on hold]

Let $f \in L^2[a,b]$. 1- In what condition(s) on a function $g:[a,b]\rightarrow [a,b]$ we can get $$f \circ g \in L^2[a,b]?$$ 2- In what condition(s) on $g:[a,b]\rightarrow [a,b]$, the operator ...
0
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1answer
16 views

Functional derivative - understanding some basics

I have the following functional $$ L[u] = \int_0^l dx [-\frac{\lambda}{2}u^2 + \frac{1}{4}u^4] = \int dx J[u]$$ Now, I need to calculate $$ \frac{\delta L}{\delta u} $$ As I understand, since I can ...
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23 views

question on functional analysis. [on hold]

Let $a$ and $b$ be arbitrary real numbers with $a<b$. Show that $[a,b]$ is closed by proving its complement is open. How do I prove this?
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1answer
31 views

doubt with topology and functional analysis [on hold]

Prove that if $x \in \mathbb R$ and $\delta(x) > 0$ in the interval $(x-\delta(x), x+ \delta(x))$ is itself an open set. How to prove this can anyone help me on this?
0
votes
2answers
33 views

Is the set of continuous functions from $[0,1] \rightarrow \mathbb{R}$ closed in the same set from $[0,1]$ to $\mathbb{C}$?

Let $X$ be the set of continuous functions from $[0,1]$ to $\mathbb{C}$, equipped with the norm $\|f\| = \int\limits_0^1 |f(x)|dx$, and let $S$ be the subspace of those functions into $\mathbb{R}$. I ...
1
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0answers
23 views

Convergence of infinite products

I wonder, parallel to the theory of summability of infinite series is there a theory for infinite products? Is there any generalized convergence method (such as Cesaro and Abel summability) for the ...
0
votes
1answer
38 views

How do I prove that Gateaux differential is linear

Reference:http://en.m.wikipedia.org/wiki/Gateaux_derivative ('Linearity and continuity' section) Let $V,W$ be Banach spaces over $\mathbb{K}$ and $E$ be open in $V$. Let $f:E\rightarrow W$ be a ...
1
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2answers
39 views

How to prove the closure $\bigcup_{n\ge 1} F_{n}$ is totally bounded and closed

Let $(X,p)$ be a metric space. Write $F$ for the set of subsets of $X$ which are closed, bounded, and non-empty. For each integer $n\geq1$, write $F_{n}$ for the set of subsets of X which are finite ...
0
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1answer
16 views

Properties of Orthonormal Systems and Projections

Let $\{e_1, ... , e_n\}$ be a finite orthonormal system in an inner product space $(E, \langle \cdot , \cdot \rangle)$, let $F :=$ span$\{e_1, ... , e_n\}$ and let $P:E \to F$ be the orthogonal ...
0
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1answer
24 views

solve equation with Intermediate value theorem…

set $a_1$,$a_2$,$a_3>0$ and $λ_3>λ_2>λ_1$ on $ℝ$. show that there are exactly two $x$’s for $a_1/(x-λ_1) + a_2/(x-λ_2) + a_3/(x-λ_3) = 0$ I tried use the intermediate value theorem but I ...
0
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1answer
29 views

Sums of projections in a C*-algebra

Let $A$ be a $C^*$-algebra, and let $p_1, \ldots, p_n \in A$ be projections, meaning $p_i = p_i^* = p_i^2$. Now assume that the sum $p = p_1 + \ldots + p_n$ is also a projection. How can one show that ...
1
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1answer
16 views

Closure of intersection with vector subspace

I am confused with the footnote on page 198 of http://www.ma.huji.ac.il/~razk/iWeb/My_Site/Teaching_files/TVS.pdf Essentially: Let $X$ be a topological vector space and $Y$ a finite-dimensional ...
1
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0answers
18 views

A question about weak convergence in Lp space [duplicate]

Suppose $1 \leq p<\infty$, given $f \in L^p (\mathbb{R})$, define $f_n (x)=n^{1/p} f(nx)$ for n=1,2,3... Prove $f_n$ converges weakly to zero in $L^p$. Now I can just know the that $ \|f_n\|_p$=$ ...
0
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0answers
7 views

Algebraic multiplicity of an eigenvalue for abstract operators

How does one define algebraic multiplicity of an eigenvalue for an abstract operator? (for a matrix the definition is clear). E.g. Consider $\partial_x^2$ on $H^2_{per}(0,1)$ then $\partial_x^2 ...
6
votes
1answer
26 views

Definition of gradient?

Reference: A primer of nonlinear analysis - Antonio & Giovanni Let $H$ be a hilbert space over $\mathbb{K}$ and $U$ be open in $H$ and $p\in U$ and $f:U\rightarrow \mathbb{K}$ be a functional ...
1
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1answer
16 views

Difference between total orthonormal set and basis

I'm learning about Hilbert spaces and related things from the book "Introductory functional analysis with applications". Now I just read the following sentence, which I don't quite understand: "A ...
1
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1answer
15 views

What is the definition of differentiation in normed space?

I'm trying to generalize implicit&inverse function theorems in Euclidean spaces to the context of Banach spaces. I'm wondering what would be the definition of differentiation in Banach space and ...
1
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0answers
27 views

Question about contractible set .

Please if i have a contractible and closed set $A$ in $X$ thene $A$ is closed and there existe a continuous function $H:[0,1]\times A\rightarrow X$ such that $H(0,u)=u, H(1,u)=p\in X.$ If i ...
1
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0answers
22 views

Does this reasoning about fourier analysis make sense?

I'm asked to show that there cannot be $\alpha_1,\alpha_2,...\in\mathbb{C}$ s.t. $$\lim_{N\to\infty}\int_{-\pi}^{\pi}|e^{it}-\sum_{k=1}^{N}a_k\sin(kt)|^2dt=0$$ Here is my attempt: Assume there are ...
1
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0answers
19 views

Bounded operator

I have a question about a operator. Let $d \in \mathbb{N}$ and $\gamma>d$. For $f \in L^{2}(\mathbb{R}^{d})$, we define $T:L^2(\mathbb{R}^{d}) \to L^{2}(\mathbb{R}^{d})$ by \begin{align*} ...
1
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1answer
24 views

Integral defined on space of matrices

I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. ...
2
votes
2answers
40 views

Self-adjoint operator- domain unique?

I was wondering about the following: Let $T : dom(T) \subset H \rightarrow H$ be a self-adjoint operator, does this mean that the domain of $T$ is uniquely defined or is it possible to make the same ...
1
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1answer
27 views

Why is this sequence relatively compact in $L^1$?

I am currently reading this paper from 1973. In short, one has given a linear continuous operator $P : L^1([0,1]) \to L^1([0,1])$ with ||P||=1 and for $f \in L^1$ a family of functions ...
2
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0answers
22 views

Existence of particular functionals in a family of linear functionals

Let $U\subset B$ be a subset of a Banach space $B$, and let $D$ be a complete topological vector space. I have given a family $\mathcal L(U)=\{L_u\ |\ u\in U\}$ of linear functionals $L_u:D\to\mathbb ...
2
votes
1answer
68 views

Can $e^{c\delta(t)}$ be rewritten some how?

Can \begin{align*} e^{c\delta(t)} \end{align*} be rewritten some how? Where $\delta(t)$ is a delta function and $c$ is some constant.
0
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1answer
23 views

Bounded Operator and p-norm (more difficult than it seems).

Let $\mathbb{R}^k$ and $\mathbb{R}$ be real vector spaces (with the usual operations of addition and scalar multiplication in each one of them) with the norm $\|\mathbf{x}\|_p$ for the space ...
3
votes
1answer
22 views

Defining a distribution

We fix the space $\mathcal{D}=\mathcal{C}^\infty_0(\mathbb{R}^n)$ as space of testfunctions. Let $(f_n)$ be a sequence of distributions with $\lim_{n\to\infty} f_n(\varphi)$ existing for all ...
0
votes
1answer
24 views

Find the orthogonal complement [on hold]

Find the orthogonal complement of a set $A = \{f \in L^2(-1,1), \int_{-1}^{1} f(x)dx = 0\}$. What does the orthogonal projection look like?
3
votes
1answer
34 views

Showing $\sum_{n=-\infty}^{\infty}\exp\left(-\pi an^2+2\pi ibn\right)=a^{-\frac{1}{2}}\sum_{m=-\infty}^{\infty}\exp\left(-\frac{\pi(m-b)^2}{a}\right)$

How do I show that \begin{align} \sum_{n=-\infty}^{\infty} \exp\left(-\pi a n^2 + 2 \pi i bn\right) = a^{-\frac{1}{2}} \sum_{m=-\infty}^{\infty} \exp\left(-\frac{\pi(m-b)^2}{a}\right) \end{align} is ...
0
votes
1answer
19 views

Show that the application $ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty})$ is not continous

I want to prove that the application $$ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty}) $$ is not continous. If I prove that this application is not bounded I have finished. So I ...
2
votes
2answers
27 views

Show that a certain norm (here $L^1$) satisfies norm properties

I hope the question title is not confusing, since my problem is actually not directly related to norms (suggestions welcome). But here we go: In some lecture example it is shown that for $f \in ...
4
votes
2answers
51 views

$f\in L^1\cap L^2$ implies $\hat f \in L^1$?

Given $f\in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$. The Riemann-Lebesgue lemma and the unitarity of the Fourier transform on $L^2$ implies that $\hat f \in L^2\cap C_0$ where $C_0$ are continuous ...
2
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0answers
31 views

n- dimensional normed linear space isomorphic to n-dimensional Euclidean space

If $(X,\|\cdot\|)$ is an n- dimensional normed linear space over R. Is it isomorphic to n-dimensional Euclidean space $R^n$. I know it is topologically isomorphic but what about isometry? I think if ...
1
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0answers
31 views

Boundedness of Helmholtz projection

Let's cosider the Helmoltz projection $P$ into solenoidal subspace defined as $$P=I-\nabla div\Delta^{-1}$$ What can I say about its boundedness as an operator in $H^s(\mathbb{R^3})$?
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1answer
25 views

Is the inverse of a causal function causal?

I am wondering if the inverse of a causal function is causal. I'll illustrate what I mean with an example: Assume $f$ is a bijection of $\mathbb R^2$ in $\mathbb R^2$. I assume $f$ is causal in the ...
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0answers
20 views

which book and document? [on hold]

I want to study about weak topology and weak star topology. So, what can I read books or documents? With 'Functional analysis', which books are good to study?
0
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0answers
24 views

weakly null sequence and a weakly Cauchy sequence

What does a weakly null sequence relate to a weakly Cauchy sequence? What did two sequence have properties ? Let $(y_n)$ be a normalized and be a $\ell^+_1$ - sequence. Assume that there are positive ...
3
votes
1answer
60 views

Invariance of the Fredholm index under finite-dimensional perturbations

Let's call a linear map $f : V \to W$ between vector spaces over some field Fredholm if $\ker(f)$ and $\mathrm{coker}(f)$ are finite-dimensional. (Equivalently, it represents an isomorphism in the ...
1
vote
1answer
31 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
0
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0answers
23 views

Spectrum of convolution operator

I was trying to find the spectrum of the convolution operator $$ J \ast u = \int_D J(x-y) u(y) dy $$ for bounded domain $D \subset \mathbb{R}$. Does anybody know it or have a reference for me? ...
1
vote
1answer
37 views

Convergence in sequence space

Let the sequence space $ s = \{ $ all sequences of complex numbers $\}$ with distance $$ d(x,y) = \sum_{j=1}^{\infty} \frac{1}{2^j} \frac{ | \xi_j - \eta_j| }{ 1 + |\xi_j - \eta_j|}.$$ Let $ x_n = ...
2
votes
2answers
27 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: ...
0
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2answers
14 views

difference between Lipschitz on any bounded set and Locally Lipschitz?

Please, what is the difference between the following expressions "Lipschitz on any bounded set" and "Locally Lipschitz"?
2
votes
1answer
52 views

Do linear operators $A$, $B$ satisfying $A = B+BAB$ commute?

I have two linear continuous operators $A$, $B$ on Banach space $X$ (for example, square matrices), satisfying the equation $$ A = B + BAB, $$ and such that the continuous inverses $(\mathrm{Id} ...
0
votes
1answer
44 views

What does $f(u)=\min!$ mean in calculus of variations?

I have a very simple notation related question. There are notes to calculus of variations [specifically: Zeidler's book "Nonlinear Functional Analysis and its Applications II/B" page 506] which states ...