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, ...

learn more… | top users | synonyms (1)

1
vote
0answers
10 views

Boundedness and Schrödinger operators

I'm working on Schrödinger operators, currently struggling with the proof of $\nabla(\Delta + i)^{-1/2}$ bounded. The domain is $H^2(\mathbb R^3)$. So, of course $(\Delta + i)^{-1}$ is bounded, and ...
1
vote
0answers
11 views

Weak solution of elliptic equation depends continuously on parameter

Suppose I have a weak formulation of the form: find $u \in H^1_0(\Omega)$ such that $$\int_\Omega b(p)(\nabla u \nabla v + \lambda uv)=0$$ holds for all $v \in H^1_0(\Omega)$ where $b:[a,b] \to ...
0
votes
0answers
11 views

Time derivative of a function involving absolute value

I have a functional looking like this $$ L[u] = \int \int_{\Omega} k \left| \nabla u \right|^2$$ , for which I want to take the time derivative $\dot{L}$. I am not sure that I handled the time ...
1
vote
0answers
4 views

Pre-dual of distributions with support in a closed subset

Usually, in order to define the space of distributions $\mathcal{D}'(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$, one considers the space $\mathcal{D}(\Omega)$ of $C^\infty$-test ...
2
votes
1answer
16 views

Counterexample Poincaré Inequality for $H_0^1$ in 2D

Is there any counterexample to the Poincaré inequality $$\int_\Omega|f|^2dx\leq C(\Omega)\int_\Omega|\nabla f|^2dx $$ for $f\in H_0^1(\Omega)$, $C(\Omega)>0$ and $\Omega\subset\mathbb{R}^2$? I ...
0
votes
1answer
16 views

Gateaux but not Frechet differentiable functional

For functional between Banach spaces X,Y: By Gateaux differentiable at $u\in X$ I mean that there exists bounded linear operator $dF(u)$ s.t. $F(u+t\xi)-F(u)=dF(u)\xi+o(t)$ for all $\xi\in X$. For ...
0
votes
1answer
17 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
0
votes
0answers
14 views

L1 norm less than BV norm

I will appreciate any hint on this Prove that if $f$ is a function of bounded variation then $\|f'\|_{L_1} \leq \|f\|_{BV}$. When $f$ is differentiable just by the Fundamental Theorem of ...
0
votes
0answers
33 views

Show that function is positive definite

I have been working on this problem for the past couple of days and I am stuck with it. It asks to show that the following function is positive definite $$\alpha(t)=\sum_{k=-\infty}^\infty ...
2
votes
1answer
25 views

Homeomorphisms of different spheres

It is pretty straightforward to show that if $X$ is a Banach space under two equivalent norms, then the respective open unit balls $B_1$ and $B_2$ are homeomorphic only by showing that $B_i$ is ...
0
votes
1answer
44 views

If two linear functionals are such that the kernel of one is contained in the kernel of the other, then they are proportional [duplicate]

Let $V$ be a vector space over $K$ and let $f,g \in V^*$ and satisfy $\ker f \subseteq \ker g$. Show there exist such $c \in K$ so that $c \cdot f =g$ How to approach this problem ?
1
vote
2answers
16 views

Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
0
votes
0answers
39 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
vote
1answer
21 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 $S=T^*?$
0
votes
0answers
34 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
vote
0answers
21 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
votes
2answers
33 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
votes
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 ...
-6
votes
0answers
25 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?
-4
votes
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
35 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
vote
0answers
29 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
39 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 ...
2
votes
2answers
43 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
votes
1answer
28 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
votes
1answer
26 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 ...
1
vote
1answer
35 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 ...
2
votes
1answer
18 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
vote
0answers
19 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
votes
0answers
8 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
vote
1answer
17 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
vote
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 ...
2
votes
0answers
31 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
vote
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
vote
0answers
20 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
vote
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
vote
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
votes
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
69 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
votes
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
23 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
52 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
votes
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
vote
0answers
32 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 $H^s(\mathbb{R^3})$?