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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).

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2answers
37 views

What does the notation $\lim_{m,n\rightarrow\infty}||f_m-f_n||_D=0$ mean and how can I prove Cauchy's criterion for uniform convergence?

$(f_n)$ is a sequence of functions which map from $D$ to $\mathbb{C}$ I want to show that $$\lim_{m,n\rightarrow\infty}||f_m-f_n||_D=0\iff||f_n-f||_D\rightarrow0$$ By Definition a sequence of ...
1
vote
1answer
16 views

Completion of inner product space

Let $(X,\langle\cdot,\cdot\rangle)$ be a real inner product space. Then, there exists a linear isometry from $X$ to its dual $X'$, whose image is dense. I know that in the case $X$ is complete, the ...
1
vote
0answers
14 views

About positive eigenvectors of $-\Delta$ on $\mathbb R^N$

Im interested whether there exists positive solutions of $$ (*)\begin{cases} -\Delta u=\lambda u\\ u\in H^1(\mathbb R^N) \end{cases} $$ for some $\lambda\in\mathbb R$. Here, we hve $-\Delta=-\sum\...
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0answers
9 views

Density in fractional Sobolev space

Suppose $s\in (0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1-\Delta)^{-s/2} L^2(\mathbb{R}^d), $$ $$ H_D^s=\{f\in H^s: f=0 \ \ a.e. on \ D^c\}. $$ Q: Is $C_c^\infty(D)$ dense in $...
0
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2answers
36 views

Fourier transform $\mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$ unbounded?

I want to prove or disprove that the Fourier transform $\mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$ is unbounded, where $\lVert\cdot \rVert_1$ denotes the $...
3
votes
1answer
28 views

critical case in $L^p$ convergence

Let $f_n$ be a bounded sequence in $L^1\cap L^{p}$ with $1<p<\infty$. Assume that $f_n\to f$ strongly in $L^1$. Then basically we have $f_n\to f$ strongly in $L^q$ for all $1\leq q<p$. But we ...
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0answers
12 views

How to prove that Frechet derivative exists and coincides with the Gateau derivative?

Let $X$ and $Y$ be Banach spaces and $U \subseteq X$ be open.Let $F:U \to Y$ be Gateaux differentiable and let the mapping $x \to F′(x)$ be continuous from $U \in L(X,Y)$. How can I prove that the ...
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0answers
15 views

Closed operator in sobolev spaces

So suppose we look at the operator $S:H_0^1(0,1) \rightarrow L^2(0,1), \ u\mapsto u' $, where $H_0^1(0,1)$ denotes the closure of the infinitely differentiable functions compactly supported in $(0,1)...
3
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1answer
26 views

Question on Banach-Alaoglu theorem: Bounded subset of a set contained in the dual space

So the Banach-Alaoglu theorem states: Let $X$ be the dual space to some Banach separable space $Z$, i.e $X=Z^*$. Take $M$ a bounded subset of $X$. Then any sequence in $M$ has a weak-* ...
2
votes
1answer
30 views

Convergence in Lebesgue-Bochner Space $L^{\infty}(0,T,L^1(\Gamma))$

Let $\Gamma$ be a compact $C^2$ manifold and suppose that $f_n$ is a non negative sequence of functions such that ${\vert \vert f_n \vert \vert}_{L^{\infty}(0,T,L^1(\Gamma))} \le C$ I am interested ...
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0answers
28 views

Continuity of $\textrm{argmin}$ set-valued mapping

Let $X \subset \mathbb{R}^n$ be a finite set and $\Phi : X \mapsto 2^{X}$ be a set-valued mapping defined as follows: $\Phi(y) := \underset{x \in X}{\textrm{argmin}} \; L(x, y)$. I'm trying to ...
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0answers
34 views

Norm of dual space of $H_0^1$

Let $H^{-1}$ denote the dual space of $H_0^1(\Omega)$. Then every $f \in H^{-1}$ can be represented as $$f(u) = \int_\Omega f^0u\ dx + \sum_{k=1}^n \int_\Omega f^k \partial_k u\ dx$$ for some ...
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0answers
25 views

approximating a decreasing function with hyperbolic functions

Let $y = f(x)$, where $x, y \in \mathbb{R}_1$ and $f \in \mathcal{C}^1$ with $f'(x) \le 0$. Partition the range using $t$ points ${y_1, \ldots, y_t}$ and - on each interval $[y_m, y_{m+1})$ - ...
2
votes
1answer
29 views

Convergence in a Hilbert Space

I have a homework question I am attempting to no avail. Let $\mathcal{H}$ be a Hilbert space. Let $X\subseteq\mathcal{H}$ be a convex set. Suppose that $(x_{n})_{n\geq 1}$ is a sequence in $X$ such ...
3
votes
1answer
43 views

Representation of linear operator between $L^p$ spaces.

I was wondering where I could find a reference to the fact that continuous linear operators: $$T:L^p(X,\mu)\to L^q(Y,\eta)$$ are of the form $T(f)(y)=\int_{X} k(x,y)f(x)d\mu$ for some $k$ satisfying ...
1
vote
2answers
35 views

Proving the uniform convergence of $\sum_{i=1}^{\infty}nx^n$ where $x \in [0,1)$ using Weirstrass M test.

I want to prove the uniform convergence of $\sum_{i=1}^{\infty}nx^n$ where $x \in [0,1)$. I know this sum has been asked a lot of times before but Im supposed to prove this with tools I have seen and ...
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0answers
37 views

Understanding a function space

I was reading a paper on Homogenization theory, where the author uses the spaces of vector valued functions. Let us consider such a space $D[\Omega; C^\infty(R^n)]$, consisting of all the compactly ...
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0answers
13 views

Partition function for nonlinear sigma model

Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. The action for the nonlinear sigma model with base $M$ and target $N$ is simply the Dirichlet energy (i.e. harmonic map energy) for maps $\phi:M\to N$: ...
2
votes
2answers
25 views

A weakly convergent sequence in a compact set, is strongly convegnet

Let $E$ be a Banach space, and $K \subset E$, compact set for the strong topology. And let $(x_n)_n$ converges for the weak topology $\sigma(E,E^*)$ to $x$. Why $(x_n)_n$ converges for the strong ...
2
votes
1answer
28 views

Operator Semigroups Simplified

How do I explain Operator Semigroups, in particular, positive operator semigroups to someone who hasn't studied math beyond high school? I just want to give a vague idea/analogy to someone to let ...
2
votes
2answers
33 views

Show that $A:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ where $A(e_n)=\lambda_ne_n$ is bounded.

Let $C\subset\mathbb C$ be closed. As $\mathbb C$ is separable then so too is the subset $C$. This means that there exists a countable subset $\{\lambda_n:n\in\mathbb N\}\subset C$ dense in $C$. In ...
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0answers
50 views

Show that $(f_n)_n$ is relatively compact in $L^p$ space

Let $I=[0,1]$, $Q=I\times I$ and $(u_n),(v_n)$ bounded sequences in $L^2(I)$. Assume $x\mapsto u_n(x), x\mapsto v(x)$ are continuous and monotone non decreasing on $I$ for all $n\in\mathbb{N}$; define\...
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1answer
64 views

7 doubts about the von Neumann algebra [on hold]

A von Neumann algebra, or $W^*$-algebra, is a $*$-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type ...
2
votes
1answer
30 views

Is the “tensoring map” from $\mathcal{H}_1 \times \mathcal{H}_2$ to $\mathcal{H}_1 \otimes \mathcal{H}_2$ a continuous map?

Let $\mathcal{H}_1$ and $\mathcal{H_2}$ two Hilbert spaces. Construct the tensor product $\mathcal{H}_1 \otimes \mathcal{H}_2$ as the set of all bounded antilinear operators from $\mathcal{H_2}$ to $\...
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votes
1answer
37 views

Convex or Concave Function

the problem is: Is the function $\phi(f)=\frac{\sqrt{f}}{2}\cdot\log f$ for $f\in L^2(\mathbb{R}_{>0})$ convex or concave. My idee or presumption is, that the function $\phi$ is concave, because ...
1
vote
1answer
37 views

$l_2(S)$is a hilbert space where S is a subset

Let S be a non-empty set and $l_2(S)$ be the set of all complex functions $f$ defined on $S$ with the following two properties: $(1) \{s:f(s)\ne0\}$ is empty or countable. $(2) \sum{|f(s)|}^2 <...
1
vote
3answers
26 views

Norm and compactness of the Operator $(Tu)(x)=\alpha(x)u'(x), u\in Y, x\in I$

Let $I=[0,1]$ and call $X$ the Banach space $C(I)$, endowed with the uniform norm. Introduce $Y=\{u\in X, u$ diffentiable on $I$ with $u'\in X\}$ and set $||u||_Y=||u||_\infty+||u'||_\infty, u\in Y, ...
5
votes
1answer
31 views

Sobolev space reflexivity problem

Let $ I $ open interval of $ \mathbb{R} $ We know that the Sobolev space $ W^{1, \infty}(I) $ is not reflexive. But, is there any easy proof of this result? Thank you in advance
3
votes
3answers
77 views

If an operator $T$ satisfy a property, then $\|Tx\|=c\|x\|$

Let $(E,\langle\cdot\;,\;\cdot\rangle)$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$. Assume that $T\in \mathcal{L}(E)$ and satisfy the following property (P)...
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0answers
39 views

Showing convexity of a set in $\mathbb{C}^k$

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert Space and $T\in\mathscr{B(\mathcal{H})}$. Suppose $$\mu_k(T):=\left\{ \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ ...
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votes
0answers
27 views

roots of concave functions

How can we show that the function $$ f(x):= x(x^{p-2}+1-p)-c(x^{p-2}+1+p) \qquad (0 < x < (p-1)^{\frac{1}{p-2}} )$$ which in it $ 1<p<2 $ and $ c=(2(p-1))^{\frac{1}{p-2}}$, has no root? We ...
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0answers
22 views

Incomplete regularized Gamma function recursive relation

I'm trying to found the detailed derivation of: $Q(\alpha + n, t) = Q(\alpha, t) + t^\alpha e^{-t} \displaystyle\sum_{k=0}^{n-1} \dfrac{t^k}{\Gamma(\alpha+k+1)}$ This is a relation necessary for the ...
3
votes
1answer
25 views

Boundedness of a linear operator

Let $X$ be a real normed linear space of all real sequences which are eventually zero with the 'sup' norm and $T:X \to X$ be a bijective linear operator defined by $$T(x_1,x_2,x_3,....)=\left(x_1,\...
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votes
1answer
24 views

Norm of Fourier series

I am reading the proof of the statement that no non-zero multiplication operator on $L^2([0,1])$ is compact in this post. And I would like to address it as a seperate post as I am only curious about ...
0
votes
1answer
23 views

Closure of $C^1[0,1]$ functions under the Lipschitz norm

I have been trying to prove that $C^1[0,1]$, i.e the space of continuously differentiable functions is closed under the $C^{0,1}[0,1]$ norm. $$||f||_{C^{0,1}}=||f||_{C^0}+\sup_{x\ne y}\frac{|f(x)-f(y)|...
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0answers
13 views

The importance of estimates of frame bounds.

A theorem contained in Christensen, Ole (1995), "A Paley-Wiener theorem for frames." Proceedings of the American Mathematical Society, 123, 2199-2201. states that Let $\{x_n\}_{n\in\mathbb Z}...
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1answer
11 views

Limit in distribution of increasing variance normal random variable

Suppose you have a sequence of normal random variables $X_n \sim N(0, n)$. Is there any random variable $X$ such that $X_n \to X$ in distribution. Here, I use the following definition: $X_n \to X$ ...
2
votes
1answer
36 views

Bounding || zx - yz || given that || x - y || < M in a Banach algebra.

Let $ X $ be an Banach algebra (not necessarily commutative), and let $ x, y, z \in X $. Suppose that $ \| x - y \| < M $. I want to bound $ \| zx - yz \| $ in terms of $ M $ by writing $ zx - yz ...
2
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0answers
35 views

Closure of finite span has non-empty interior

Be $X$ a Banach space with countable basis. Suppose $(x_n)_{n \in \mathbb{N}} $ is a sequence in $X$ (allowing repetitions), such that $$(\forall x \in X)(\exists M_x \subseteq \mathbb{N})\left(|M_x|...
1
vote
1answer
27 views

Dualization map is surjective

I am practicing for my exam and I want to solve the following problem. Let $X,Y$ be normed reflexive spaces. Show that the "Dualization map" $':B(X,Y)\to B(Y',X')$, $T\mapsto T'$ is surjective I ...
1
vote
1answer
41 views

Fréchet Derivative of functionals which depend on the derivative

Let $\Omega:=[x_a,x_b]\times[t_a,t_b]\subset \mathbb{R}^2$ and consider the functional $D:C^2\left(\Omega,\mathbb{R}\right)\rightarrow \mathbb{R}$ such that $$D[y]= \iint_{\Omega}d(x,t,y) \, dx \, dt$$...
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0answers
19 views

Classical Shearlet system forms a Parseval frame.

I have been reading about shearlets & frames and there is a part that I'd like help with. I have only studied the basics of functional analysis and wavelets. First few definitions: Classical ...
1
vote
1answer
43 views

Is $i: (C^1, ||·||_{W^{1,2}}) → (C^0, ||·||_∞)$ a linear, continuous, compact map?

Consider the map $$i: (C^1[0,1], ||·||_{W^{1,2}}) → (C^0[0,1], ||·||_∞)$$ which maps every function to itself, and with Sobolev norm defined as $$||u||_{W^{1,2}}=||u||_{L^2}+||u'||_{L^2}.$$ ...
1
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0answers
19 views

Non expansive function

Let $X$ be a Banach space,$F:D \subset X \rightarrow X$ and $\lambda >0$. If $F+\lambda I $ is onto then $R_{\lambda}=(I+\lambda F)^{-1} $ is non expansive, i.e, $|R_{\lambda}(x) -R_{\lambda }(y) | ...
2
votes
1answer
34 views

A sequence of bounded $C^1$ functions whose derivatives are unbounded.

What is an example of a sequence of functions, $(f_n)_{n=1}^\infty\subset C^1([a,b])$, which are bounded in $C^1([a,b])$ under $\|\cdot\|_{\infty}$ but are such that their first derivatives $\|f'_n\|_{...
0
votes
1answer
19 views

Regarding equivalent conditions of Frechet differentiability

Gateaux and Frechet differentiability in a Banach space are defined as below. Can you tell below how (ii) implies (i). The rest is easy.
0
votes
3answers
30 views

orthonormal basis of infinite dimensional Hilbert space H is not a basis of H as vector space?

Apparently the orthonormal basis $(e_n)_{n\in \mathbb{N}}$ of the Hilbert space $H$ (in special case, infinitly dimensional) is not a basis of $H$ as a vectorspace. Is there a way to prove this?
0
votes
1answer
27 views

$\frac{d}{dx}:C^1([0,1])\to C([0,1])$ as a closed, unbounded operator

First recall that a (potentially unbounded) operator $T:D(T)\subseteq X\to Y$ is closed whenever $(x_n)_{n=1}^\infty\subset D(T)$ convergent to $x\in X_0$ and $(Tx_n)_{n=1}^\infty$ convergent in $X_1$ ...
0
votes
0answers
21 views

Weak convergence and convergence in norm implies convergences in a locally uniformly convex Banach space. [duplicate]

Let $X$ be a locally uniformly convex Banach space, then if $x_n \overset{w}{\rightarrow}x $ and $|x_n| \rightarrow |x| $ implies $x_n \rightarrow x$. $X$ is locally uniformly convex, Does the same ...
0
votes
1answer
16 views

Supnorm Defined on set of all uniformly continuous functions is Banach space

If we define Supnorm on $C(\bar{U})$ is $$||f||_{C(\bar{U})}:=\sup_{x\in U} |f(x)|.$$ on $C(\bar{U})$ is Banach space what I know is : I'm proving this normed space since $||f||_{C(\bar{U})}=\...