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).
49,283
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Metric space $l_p$ which is closed and bound and not compact?
Consider the space $l_p =\{(x_n): \sum_{n=1}^\infty \vert x_n\vert^p < \infty\}$ with the norm $\|.\|_p$.
Take: $A$ of all members in the sequence $(u_n)_n \subset l_p$ such that $u_n = (0,...,0,1,...
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If $f \in L^1 (\mathbb R^d)$ then $\lim_{h \to 0} \int |f(x+h)-f(x)| d x = 0$
A sequence of mollifiers $(\rho_n)_{n \geq 1}$ is any sequence of functions on $\mathbb{R}^d$ such that
$$
\rho_n \in C_c^{\infty} (\mathbb{R}^d), \quad \operatorname{supp} \rho_n \subset \overline{B(...
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sufficient condition for a finite dimensional normed real vector space to be a hilbert space [closed]
This comes from Exercise 1.2.24 from the book: A course in metric geometry by Dmitri Burago, Et al.
Let $V$ be a finite dimensional normed real vector space, if for any $x,y\in V$ with the same norm, ...
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2
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How can I find a lift for this circle homeomorphism?
Let me consider the circle homeomorphism $T:\Bbb{R}/\Bbb{Z}\rightarrow \Bbb{R}/\Bbb{Z}$s.t. $T(x)=x+\frac{1}{2}$ if $x\in [0, 1/2]$ and $T(x)=x-\frac{1}{2} $ if $x\in [1/2,1]$.
Now I want to find a ...
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1
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23
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Brezis's Theorem 4.26: how to obtain $\|\rho_n \star f\|_{L^\infty (\mathbb{R}^N)} \le C_n\|f\|_{L^p(\mathbb{R}^N)}$?
A sequence of mollifiers $\left(\rho_n\right)_{n \geq 1}$ is any sequence of functions on $\mathbb{R}^N$ such that
$$
\rho_n \in C_c^{\infty}\left(\mathbb{R}^N\right), \quad \operatorname{supp} \rho_n ...
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1
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Bound on $\|ABv\|$ in terms of $\|Av\|$
I have a Hilbert space $H$ with:
An unbounded self-adjoint operator $T$.
Another unbounded operator $S$ such that $S=PTP^{-1}$, where $P$ is a bounded invertible operator (not necessarily unitary).
...
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1
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0
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14
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Folland Chapter 5 Exercise 39
The problem:
$X,Y,Z$ are Banach spaces and $B:X\times Y\to Z$ is a bilinear map which is separately continuous. i.e. $\forall x\in X$ the map $B(x,\cdot):Y\to Z$ is continuous, and $\forall y\in Y$ ...
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1
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Does strictly positive-definite imply local minimum?
Let $\mathcal H$ be a Hilbert space and $f: \mathcal H \to \mathbb R$ a $ C^2$ function. Suppose $z \in \mathcal H$ is a critical point of $f$,
and the Hessian of $f$ is strictly positive at $z$.
Can ...
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2
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71
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Difference in usage between function, mapping, functional, form, and operator?
The word function has many synonyms (or close to synonyms), including:
map
functional
form
operator
transformation
What is the difference, in meaning or usage, between them? I understand that exact ...
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0
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12
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Question on finding a Hilbert space isomorphism between $L^2([0,R]^n)\otimes L^2([0,R]^m)$ and $L^2([0,R]^{n+m})$
Let $R > 0, n,m\in\mathbb{N}$ and suppose that we have already showed that the map $f:L^2([0,R]^n)\otimes L^2([0,R]^m)\to L^2([0,R]^{n+m})$ is a linear bijective isometry between the basis vectors ...
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Density properties of variable-order fractional Sobolev spaces $W^{s(\cdot,\cdot),p}(\mathbb{R}^N)$
Background: My research focuses on studying variable-order fractional $p$-Laplacian evolution equations, which relies on Theorem 3.4 presented in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-...
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$\mathcal C_c^\infty (\Omega)$ is dense in $L^\infty (\Omega)$ w.r.t. the weak topology $\sigma(L^\infty, L^1)$
I'm trying to prove a result mentioned in this thread, i.e.,
Let $\Omega$ be an open subset of $\mathbb R^d$. Then $\mathcal C_c^\infty (\Omega)$ is dense in $L^\infty (\Omega)$ w.r.t. the weak ...
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0
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23
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Corollary 4.24 in Brezis's Functional Analysis
I'm trying to prove below result in Brezis's Functional Analysis, i.e.,
Corollary 4.24. Let $\Omega$ be an open subset of $\mathbb R^d$. Let $u \in L_{\text{loc}}^1 (\Omega)$ such that
$$
\int_{\...
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0
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Assume that $T$ is positive, prove that $\|T\|$ equals the largest singular value of $T$.
Let $T$ a linear operator between finite-dimensional inner product spaces. Let $\|T\|$ be the operator of norm. Assume that $T$ is positive, prove that $\|T\|$ equals the largest singular value of $T$....
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1
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Minkowski functional of a closed convex set [closed]
Let $A$ be a closed convex set with $0$ in its interior and $p_A$ be its Minkowski functional. I'm trying to prove that for any $c \ge 0$, $\{x \in X : p_A(x) \le c\} = cA$ but I'm confused about the ...
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1
answer
42
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Group von Neumann algebras and the $\|\cdot\|_{2}$ norm
Let $G$ be a discrete group and $LG$ the corresponding group von Neumann algebra. For a set $F\subset G$, let $P_{F}:\ell^{2}(G)\rightarrow\ell^{2}(F)$ denote the orthogonal projection onto $\ell^{2}(...
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0
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If $P$ is self-adjoint and $P^2$ is a projection, when is $P$ a projection?
Let $H$ be a Hilbert space and $P:H \to H$ a linear operator.
I am aware that if $P$ is self-adjoint and idempotent then it's a projection.
My question is: if $P$ is self-adjoint and $P^2$ is a ...
0
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0
answers
15
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Sobolev space on Carnot group with zero trace
Let $\mathbb{G}=(\mathbb{R}^{n},\circ)$ be a Lie group on $\mathbb{R}^n$ and $\mathfrak{g}$ be the corresponding Lie algebra of $\mathbb{G}$. Let $X_{1},\ldots,X_{m}$ be the left-invariant smooth ...
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0
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18
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Convergence of the sequence of the unit vectors w.r.t. 2-Norm
If we consider the space of sequences $\ell_2$, as well es the sequence of unit vectors $ (e_n)_n $, does $(e_n)_n$ converges to zero vector with respect to $\left\| .\right\|_2$?
My answer would be ...
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1
answer
51
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Rudin's RCA The Hahn-Banach Theorem.
There is the theorem:
If $M$ is a subspace of a normed linear space $X$ and if $f$ is a bounded linear functional on $M$, then $f$ can be extended to a bounded linear functional $F$ on $X$ so that $||...
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1
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1
answer
43
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Why is the wiener algebra a subset of continuous functions
I am studying Fourier analysis and I am currently reading about the Wiener algebra. The Wikipedia page claims that $A(\mathbb{T}) \subset C(\mathbb{T})$ (see: https://en.wikipedia.org/wiki/...
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1
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29
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Is the canonical map $T : L^1(\Omega, \mu, \mathbb R) \to (L^\infty(\Omega, \mu, \mathbb R))^*$ injective?
Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. We define a map
$$
T : L^1(\Omega, \mu, \mathbb R) \to (L^\infty(\Omega, \mu, \mathbb R))^*
$$
by
$$
(T u) (f) := \int_{\Omega} uf \ ...
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1
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How can you show for an infinite set of basis that it spans the vector space?
So for a finite set of vectors it is important to show that it spans the set and that the vectors are linearly dependent.
But for a vector space with infinite dimensions, how can you show that a set ...
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0
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25
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Ultracontractive semigroup generated by the fractional Laplacian
Let $A=-(-\Delta)^s$ with domain $D(A)=$ {$\phi\in\mathbb{H}_{0}^{s}(\Omega),(-\Delta)^s\phi\in L^2(\Omega)$ }. $(-\Delta)^s$ is the fractional Laplacian of order $s\in(0,1)$.
We know that $A$ ...
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0
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63
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$\ker(f)=\ker(g)$ implies $f=cg$ [duplicate]
Let $X$ be a normed space, and $f, g$ be linear functionals on $X$ such that they have the same kernels. I have to prove that there exists some scalar $c$ such that $f=cg$.
If $f$ and $g$ are zero ...
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0
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1
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Multiplicativity of order relation for self-adjoint operators
Let $A$ be a $C^*$-algebra. It is known that the order relation on the set of self-adjoint elements of $A$ is well-behaved with respect to addition, i.e. if $a\leq b$ and $c\leq d$ then $a+c\leq b+d$.
...
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1
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Is there a normed space that it has a element (except 0) that every functional of the dual of the normed space mapping it to 0?
Suppose $X$ is a normed vector space, $X'$ is its dual space. My question is is there a $X$ satisfied the following condition.
There is a element $x$ in $X$ which is not $0$ and it satisfied that ...
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0
answers
49
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Prove that balls are not convex in $C(\mathbb{R})$.
I must solve this problem from Rudin's Functional Analysis book. Here $C(\mathbb{R})$ is the space of continuous complex-valued functions with real domain.
I've tried giving many examples of functions ...
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0
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In the proof of Neural Tangent Kernel stays constant in infinite width limit, why the norm of the dual mapping operator equals operator norm of kernel
For a fixed distribution $p^{in}$ on the input space $ \mathbb{R}^{n_0}$,
consider a function space $\mathcal{F}$ defined as $\{{f: \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_L}}\}$.
On this space, ...
1
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1
answer
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How to calculate the convolution product of $(H_0 e^{\alpha t}) * (H_0 e^{-\alpha t})$?
everyone!
I am doing an exercise concerning the resolution of differential eaqtion in the sense of distributions.
Let $\alpha \in \mathbb{R}_+^*$. Let be the differential equation, for $f \in \mathrm{...
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2
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0
answers
23
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Riemann-Liouville Fractional Integral operator in $L^p(a,b)$ : finding the operator norm
I've recently come across the Riemann-Liouville operator in $L^p(a,b)$ (with $1\leq p \leq +\infty$), whose definition is :
$V^\alpha
> f(t)=\dfrac{1}{\Gamma(\alpha)}\int_a^t(t-x)^{\alpha-1}f(x)dx$...
1
vote
1
answer
29
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Rudin's RCA $5.17$ proposition
There is the proposition:
Let $V$ be a complex vector space.
(a) If $u$ is the real part of a complex-linear functional $f$ on $V$, then
$$f(x)=u(x)-iu(ix)$$ for all $x\in V$.
There is the proof:
If $\...
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0
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Realization of the Fractional Laplacian Operator in $L^2(\Omega)$.
Consider $\Omega \subset \mathbb{R}^N$ a bounded and smooth domain. Moreover, consider that $u(x) = 0$ for all $x \in \mathbb{R}^N \backslash \Omega$. It is well known that Laplace's fractional ...
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Metric on bounded sequences that induces pointwise convergence
I got the following question in my functional analysis course.
let B($\mathbb{N}$) be the linear space of bounded functions $f:\mathbb{N}\rightarrow\mathbb{R}$ (i.e bounded sequences). Find a metric d ...
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1
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0
answers
19
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Counterexample of Sobolev Embedding Theorem $W^{1,n}(\mathbb{R}^n) \nsubseteq L^\infty (\mathbb{R}^n)$ [duplicate]
I am looking for a counterexample of Sobolev Embedding Theorem $W^{1,n}(\mathbb{R}^n) \nsubseteq L^\infty (\mathbb{R}^n)$.
I feel that somehow the log(log()) function could provide an example but I am ...
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0
answers
83
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Most cited areas in Pure Mathematics [closed]
which areas of Pure Mathematics are the most cited? I cannot find any concrete data for this; all I can find are the most cited mathematicians, not the most cited fields. It appears as though Number ...
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0
answers
25
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Norm of Hermite polynomials in a non-standard Gaussian
The Hermite polynomials I am using satisfy the recurrence $H_k'(x) = kH_{k-1}(x)$ and they satisfy the property if $d\mu_{1/2} = (2\pi)^{-1/2}e^{-x^2/2}$ is the standard Gaussian then $||H_k||_{L^2(\...
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1
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The closure of the set of linear combinations
The vector space generated by a subset $A$ of a normed space X is the set of finite linear combinations of its elements, and will be denoted by $Lin(A)$.
$$Lin(A)=\{\sum^{n}_{j=1}\lambda_{j} a_{j}: \...
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3
votes
1
answer
26
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bounded normal operator and spectrum
Problem: If A is a bounded normal operator, the spectrum $\sigma(A)=\{s+it:s \in \sigma(B),t \in \sigma(C)\}$, where B, C are bounded self adjoint operators which commute.
Fact: A bounded normal ...
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1
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0
answers
9
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Singularity in the Fractional Laplacian definition
It is well known that Laplace's fractional operator is defined by
$$(-\Delta)^s u(x) = c_{n,s}\lim_{\varepsilon \to 0^+}\int_{\mathbb{R}^N \backslash B_\varepsilon (x)} \frac{u(x) - u(y)}{|x - y|^{N + ...
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2
votes
0
answers
57
views
Estimating the norm of a functional in $W^{-1,2}$
Suppose we have $u,v$ satisfying a ‘transport’ equation on $[0,1] \times [0,T]$; $$\partial_{t}u +v\partial_{x}u =0.$$
We assume further that $v \in L^{2}(0,T; L^{2})$, $\partial_{x}u \in L^{\infty}(0,...
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vote
1
answer
44
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Idea about the projections of the $C^*$-algebra $pAp$, where $p$ is a projection in $A$
Let $A$ be a unital $C^*$-algebra and $p$ be a projection in $A$. Now consider the $C^*$-algebra $pAp$. I want to classify the projections in $pAp.$ I know that, if $q \in A$ is a projection such that ...
- 1,777
1
vote
0
answers
20
views
Why is $df_x(h_1+h_2)=df_x(h_1)+df_x(h_2)$ when $f(x+y)\le f(x)+f(y)$ and $f(\lambda x)=\lambda f(x)$ for $\lambda\ge 0$?
Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is a function satisfying $f(x+y)\le f(x)+f(y)$ (convex) and $f(\lambda x)=\lambda f(x)$ (positive homogenous) for all $t\ge 0$, all $x,y\in\mathbb{R}^n$.
Fix an $...
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3
votes
2
answers
128
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Does the operator have any eigenvalues, also is it compact?
Question:
Let $H=L^2([0,1],m)$, where $m$ is the Lebesgue measure, and consider th e operators $M,S\in L(H,H)$ given by
$$ Mf(t)=tf(t), \; Sf(t)=f(1-t), f\in H, t\in [0,1]$$
You are not asked to ...
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1
vote
0
answers
59
views
Proposition 4.20 in Brezis's Functional Analysis
If $f \in \mathcal C^k (\mathbb R^n)$ and $\alpha = (\alpha_1, \ldots, \alpha_n)$ is a multi-index of length $|\alpha | = \alpha_1 + \cdots + \alpha_n \le k$, then we write
$$
D^\alpha f := D_1^{\...
- 13.7k
2
votes
2
answers
58
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Subspace of a Banach Space that is neither dense nor closed
Does anyone know any basic examples of subspaces of Banach spaces that are neither dense in that space, nor closed? The kernel of any continuous functional is closed, and the kernel of any ...
- 3,599
0
votes
0
answers
58
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How to prove that a subspace is weakly$^*$ closed
Let $ba:=ba(\Omega,\mathcal{F})$ the space of all finitely
additive set functions $\mu: \mathcal{F} \to \mathbb{R}$ with finite total variation $\|\cdot\|_{var}$ where
$$\|\mu\|_{var}:=\sup\left\{\...
- 57
2
votes
1
answer
91
views
Bergman Kernel for $L_p$ space (p $\neq$ 2)
For a given space $\mathcal{X}$, consider the Banach space $L_p(\mathcal{X},\mu)$ for the measure space $(\mathcal{X},\mu)$. I'm trying to understand the Bergman kernel for $L_p$ when $p \neq 2$.
For $...
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0
votes
1
answer
43
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An incomplete metric space.
I didn't understand the last part, would someone be kind enough to explain it to me? More precisely, because since $x$ is continuous we can conclude that
$$x(t)=0\quad\text{if}\; t\in \left[0,\frac{1}{...
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1
answer
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is my proof, that the length of any continuously differentiable curve is finite, rigorous?
I have a proof but im not sure that it is 100% rigorous.
I started with the defnition of the length of a curve:
Integral from a to b over ||y´(t)|| dt is more or equal then
|| Integral from a to b y´(...
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