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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25 views

Question about the proof $X'$ reflexive $\Rightarrow X$ reflexive.

I have a doubt in the proof I have been given of the fact: For a Banach space $X$, if $X'$ is reflexive then $X$ is reflexive. This is proven by showing first theorem 1 and theorem 2, which I quote ...
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2answers
44 views

What is the divergence of a distribution?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ If $p\in \mathcal D'(\Omega)$, then $$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\...
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0answers
23 views

Dimension and separability of $\ell^2(I)$?

Let $\ell^2(I)= \left\{ x:I\rightarrow \mathbb C\mid \sum _{i\in I} |x_i|^2<\infty \right\} $ with inner product $\sum_{i\in I}x_i \bar y_i$. I am supposed to find the dimension of this space and ...
2
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1answer
99 views

Advanced Linear Algebra vs Functional Analysis

I have a couple questions regarding Advanced Linear Algebra vs Functional Analysis. 1) Do these courses help in understanding or have applications in: Machine Learning Quantitative Finance, ...
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1answer
49 views

Definition of the Laplacian as an operator from $H_0^1(\Omega)$ to $H_0^1(\Omega)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $f\in L^2(\Omega)$ and $$\langle f\rangle:=\left.\langle\;\cdot\;,f\rangle_{L^2(\Omega)}\right|_{\...
2
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1answer
48 views

Weak convergence of product in $L^2$

Let $f_k\in L^\infty[0,1]$ and $g_k\in L^2[0,1]$ be two sequences such that $f_k\to f$ a.e., $\left\|f_k\right\|_{L^\infty}\leq c$ for any $k$ and $g_k\rightarrow g$ weakly in $L^2[0,1]$. Why does $...
2
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1answer
41 views

Norm of the Resolvent

Let $\mathbb{H}$ be a Hilbert space, $A$ a self-adjoint operator with domain $D_{A}$, $R_{A}$ the resolvent of $A$, and $z$ a point in the resolvent set $\rho(A)$. How could you prove the inequality \...
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2answers
33 views

Resolvent of a Self-Adjoint Operator

Let $\mathbb{H}$ be a Hilbert space and $A$ a self-adjoint operator with domain $D(A) \subset \mathbb{H}$.. Suppose that the spectrum $\sigma(A)$ of $A$ is contained in $[0,\infty)$. Let $R_{A}(z)$ be ...
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1answer
41 views

Prove the completion of the span of $ \left\{ e^{i\lambda t} \right\} _{\lambda\in \mathbb R}$ is not separable

Let $G$ be the span of $ \left\{ e^{i\lambda t} \right\} _{\lambda\in \mathbb R}$ with inner product $$ \left\langle f,g \right\rangle =\lim _{T\rightarrow \infty}\frac 1{2T}\int_{-T}^Tf\bar g .$$ I ...
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24 views

Does the Parseval identity imply the completeness of an orthonormal system?

Let $V$ be an inner product space which is not complete. Can there by an orthonormal system satisfying the Parseval identity for each vector but which is not complete i.e the only vector orthogonal to ...
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1answer
35 views

Prove $\exists x$ such that $\|Ax-b\|$ is minimal and this is unique if $A$ is invertible

Suppose $A\in \mathbb R^{m\times n}$. I'm supposed to prove $\exists x$ such that $\|Ax-b\|$ in minimal and this $x$ is unique if $A$ is invertible, in which case I also need to exhibit a formula. I ...
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1answer
33 views

Find $\inf _{a,b,c\in \mathbb C}\int _{[-\pi,\pi]}|x+a+b\cos x+c\sin x | ^2dx$

I need to find $\inf _{a,b,c\in \mathbb C}\int _{[-\pi,\pi]}|x+a+b\cos x+c\sin x | ^2dx$. I thought about using the fact $\int _{[-\pi,\pi]}f\bar g$ is an inner product $ \left\langle f,g \right\...
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1answer
16 views

Metric projection onto nonnegative sequences

I need to find the metric projection onto non-negative sequences in $\ell^2$. Intuitively I'm thinking each element in the sequence should be sent to the maximum between it and zero, since that should ...
1
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1answer
55 views

Is $X^{***}$ useful in banach space theory?

I was wondering why i have never seen $X^{***}$ be used anywhere in functional analysis. I know that $X$ can be viewed as a subspace of $X^{**}$, and in this way you can identify a subset $C$ of $X^{*...
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0answers
13 views

Bound of mollified in $H^{-2}$

Let $f\in L^2((0,T); H^2) $ with $ \partial_t f \in L^{2}((0,T);H^{-2}) $ and let $ \eta_{\varepsilon} $ a standard mollifier sequence in $ (t,x) $, then there exists a constant $ C $ independent of $ ...
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2answers
61 views

Limit of partial sums: $\lim_{n\to\infty} \frac1n\sum_{k=1}^n f(k)=0$ if $\lim_{k\rightarrow\infty}f(k)=0$ [duplicate]

I want to argue that $$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n f(k)=0~~~~~~~ {\rm if}~~~~~ \lim_{k\rightarrow\infty}f(k)=0.$$ This identity does not seem to hold always, but seems to hold ...
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1answer
36 views

aggregate two quadratic functions

I have a quadratic function$$W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j.$$ Denote the input vector as $\textbf{x}$, in quadratic form, $W(\textbf{x})=\textbf{x}^TM\textbf{x}$, where $...
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88 views

Spectacular failure of Lebesgue differentiation for rectangles

Let $\mathcal{R}$ be the set of rectangles in the plane and, given $f \in L^1$ let $$ f^*(x) = \sup_{x \in R \in \mathcal{R}} \frac{1}{ \lvert R \rvert} \int_R \lvert \, f \,\rvert $$ as defined in ...
6
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1answer
53 views

If B(X) is isomorphic to B(Y), does that mean X is isomorphic to Y (for X and Y Banach spaces)?

Let $X$ and $Y$ be Banach spaces such that $\mathcal{B}(X)$ is linearly isomorphic to $\mathcal{B}(Y)$ (where $\mathcal{B}(\cdot)$ denotes the algebra of bounded linear operators). Must it always be ...
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13 views

Example of strongly continuous unitary semigroup where Trotter limit does not exist

I am looking for an example of two strongly continuous unitary semigroups $U(t),V(t)$ for which the Trotter limit $[U(t/n)V(t/n)]^n \phi$ does not exist for some $t,\phi$.
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1answer
17 views

Inequality on embedding between spaces

let $\Omega$ be an open set. We are known that $L^2(\Omega) \hookrightarrow H^{-1}(\Omega)$, so for all $f \in L^2(\Omega)$, we have $\|f\|_{H^{-1}} \lesssim \|f\|_{L^2}$. If we assume $f$ is ...
2
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0answers
31 views

Sup of a linear function

Let $X$ be a banach space or simply a normed space and $C$ a convex (closed) subset of $X$. It is true that if $x \in C$ is such that $f(x)=\sup f(C)$, (in other words $x$ is a supporting point for $C$...
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1answer
30 views

If $L$ is a continuous linear form on a dense subspace of a Hilbert space $H$, what do we mean by the claim $L\in H$?

Let $H$ be a $\mathbb R$-Hilbert space $D(\mathfrak a)$ be a dense subspace of $H$ $\mathfrak a:D(\mathfrak a)\times D(\mathfrak a)\to\mathbb R$ be bounded, i.e. $\exists c\ge 0$ with $$\left|\...
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0answers
28 views

Estimate the Sobolev norm of negative order of a function

I would like to estimate the Sobolev norm of order $-1$ of the function $f(x)$ which is defined as follows: Let $\psi\in C^\infty_c(\mathbb{R})$ be a compactly supported smooth function on the real ...
0
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2answers
30 views

$\mathcal{L}(\ell_2,\ell_2)$ is not separable and a isomerty $T\colon\ell_\infty\to\mathcal{L}(\ell_2,\ell_2)$

I have to prove that the operator $T\colon\ell_\infty\to\mathcal{L}(\ell_2,\ell_2)$ such $T((a_j)_{j=1}^{\infty})((b_j)_{j=1}^{\infty})=(a_jb_j)_{j=1}^{\infty}$ is an linear isometry. How can I show ...
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16 views

Is the norm of tensor fields just Hilbert-Schmidt norm/ generalized $L^p$-norm?

As I am rather comfortable with functional analysis language and was new to Riemannian geometry, I am curious when inspecting the norm of Ricci tensor, which is: $$|\mathrm{Ric}|^2=g^{ij}g^{kl}R_{ik}...
3
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1answer
43 views

Why is the Maximum in the Min-Max Principle for Self-Adjoint Operators attained?

Let's consider a self-adjoint operator $A$ (not necessarily bounded) on a Hilbert space which is bounded from below, with domain $D$ and whose resolvent is compact. Then, the spectrum consists solely ...
3
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2answers
47 views

Pointwise convergence of holomorphic functions

Let $(g_n)_n$ be a sequence of holomorphic functions on $U$, where $U$ is the open unit disk. Suppose the first $k$ derivatives of $g_k$ at zero all vanish, $g_k(0) = 0$, and finally that $g_n$ ...
2
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0answers
27 views

Convergence of Compact Linear Opeators in $L^2([0,1])$

Let $A_n: L^2([0,1]) \to L^2([0,1])$ be $(A_nf)(x)= \int_0^1 sin(n\pi(x-y))f(y)dy$. Are they compact operators? Is there any kind of convergence? They are continous since $\|A_nf\| \leq \|f\| $. ...
2
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1answer
38 views

Second adjoint of the canonical embedding

Suppose that $X$ is a Banach space. Denote by $\kappa_X$ the canonical embedding of $X$ into $X^{**}$. Do we always have $$(\kappa_X)^{**} = \kappa_{X^{**}}? $$
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1answer
35 views

What is a “functional model” in the context of model operators?

I'm reading a survey paper on model spaces, there is written "the study can therefore be restricted to CNU operators, which with some additional hypotheses obtain concrete a functional model" What is ...
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1answer
28 views

Can we talk about the adjoint of a linear operator defined on a distribution space?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ and $$...
0
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0answers
29 views

A question about $L^\infty$

If we have a funtion $f:\mathbb{T}\to H$, $\sum_{k\in\mathbb{Z}} x_ke^{ikt}$, where $\mathbb{T}$ is the torus and $H$ is a Hilbert space: What is the relation between this function belonging to $L^\...
0
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0answers
35 views

precise description of maps preserving locally equal functions

Let S be a map of $C^\infty(X)$ onto itself, where $C^\infty(X)$ stands for the space of smooth functions on a smooth manifold $X$. Suppose that $Sf = Sg$ near a point y whenever $f=g $ near a point $...
2
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1answer
36 views

Proof using Banach fixed point theorem

Theorem.$\ $ Let $E$, $F$ be two Banach spaces, $U$ an open ball in $E$, $V$ an open ball in $F$ of center $y_0$ and radius $\beta$, and $v$ a continuous mapping of $U \times V$ into $F$. Suppose that ...
1
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1answer
29 views

Verifying reproducing property of common kernels in RKHS

Consider a reproducing kernel Hilbert space $\mathcal{H}$, whose elements are functions from $X \rightarrow \mathbb{R}$. Let $t \in X$. Since $\mathcal{H}$ is a RHKS, there exists a unique function $...
2
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0answers
31 views

Metrization unit ball in $X^*$ and Urysohn's Theorem

Evening, guys! I'm looking for applications of the Urysohn Metrization Theorem. Well, My first thought was prove that the unit ball in the dual, with $w^*$-topology, of a separable Banach space $X$ ...
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0answers
28 views

Is $\exp \left(-\sum_{i=1}^d \frac{(x_i - y_i)^2}{s_i^2} \right) $ analytic in $\mathbf{x}$ on $\mathbb{R}^d$?

I would like to know whether the following statement is true. Conjecture: For all $\mathbf{y} \in \mathbb{R}^d$, $s_1>0,\ldots,s_d>0$, $f_{\mathbf{y}}(\mathbf{x}) := \exp \left(-\sum_{i=1}^d \...
0
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1answer
31 views

Smooth Approximations in $L^2((0,1))$

Let $L^2((0,1))$ be as usual the Lebesgue space of measurable complex-valued functions $f:(0,1) \rightarrow \mathbb{C}$ such that $\int |f(x)|^2 dx < \infty$. It is a well known fact (see e.g Lieb ...
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22 views

A $C^1$ function in Orlicz Sobolev space

How to prove that thi functional is $C^1$: $$ I(u)=\int_{\mathbb{R}^N} \Phi(|\nabla u|)+\Phi(|u|) dx-\int_{\mathbb{R}^N} F(u) dx $$ Where $\Phi$ is an N-function and $F(t)=\int_{0}^t f(s) ds$ where ...
1
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2answers
56 views

A corollary of the Hahn-Banach theorem

Let $Z$ be a subspace of normed linear space $X$ and that $y$ is an element of $X$ whose distance from $Z$ is $d$. Then there exists a $\Lambda \in X^* $ (the dual space of $X$) so that $\| \Lambda\| \...
2
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0answers
41 views

Continuity and measurability of differential operator [closed]

Let $\mathcal{C}^k(\mathbb{R},\mathbb{R}^n)$ denote the space of $k$-times differentiable $\mathbb{R}^n$-valued functions on $\mathbb{R}$ equipped with the topology of uniform convergence of ...
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0answers
35 views

Complex version of Lax-Milgram Theorem

I'm trying to prove Lax-Milgram Theorem in the complex case, i.e. Let $X$ be complex Hilbert space and let $f\in X'$, its topological dual. If $a(\cdot,\cdot):X\times X\to \mathbb{C}$ is ...
1
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1answer
99 views

Measurable Functions with Common Sigma Sub-Algebras

Let $X:\mathbb{R}\rightarrow\mathbb{R}$ be a non-constant function, measurable with respect to the Borel-Algebra $\mathcal{B}$ and $\sigma(X)$ the sigma-algebra generated by $X$. Let $\mathcal{A}\...
3
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0answers
43 views

Does strong positivity of T implies strong positivity of T*

Let $X$ be a Hilbert space with a positive self dual cone $K$. Then a bounded operator $T$ is called positive if $T(K) \subseteq K$ Strongly positive if $T\left(\mathrm{int}\left(K\right)\right) \...
0
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1answer
43 views

Inner product on a vector-valued function space

A possible inner product on a function space $F=\{f:\mathbb{R}\to\mathbb{R}\}$ $$\int f_1(x)f_2(x)dx$$ Think of a space $F=\{f:\mathbb{R}^n\to\mathbb{R}\}$. How might I define an inner product on ...
2
votes
1answer
47 views

How to show that a Schwartz distribution is in a Lebesgue or Sobolev space?

It is known that all $L^p$ spaces (and, consequently, all $W^{s,p}$ spaces) can be embedded in the space of Schwartz distributions $\mathcal D '$. There is a problem, though: how do I check whether ...
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0answers
30 views

Prove that $\left.F\right|_{\left\{ϕ∈C_c^∞(Ω,ℝ^d):∇⋅ϕ=0\right\}}=0⇔∃p∈C_c^∞(Ω)$ with $F=∇p$, for all $F∈H_0^1(Ω,ℝ^d)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$, $$H:=\...
1
vote
3answers
56 views

Bounded linear operator, strange definition.

Let $L:X\to Y$ an linear operator. I saw that $L$ is bounded if $$\|Lu\|_Y\leq C\|u\|_X$$ for a suitable $C>0$. This definition looks really weird to me since such application is in fact not ...
1
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1answer
25 views

I'm confused of the test problem of Weak Derivatives

But why the problem says that "prove it" with such special $\varphi$?