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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1answer
25 views

Spectrum of the Laplacian for free space versus spectrum of Laplacian with boundary conditions?

In this question the spectrum of the Laplacian in free space is defined as $]-\infty, 0]$. So it seems the spectrum can be determined without regard to boundary conditions? If instead we consider the ...
1
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1answer
23 views

How to think of Sobolev spaces $W^{k, p}$ for a function that is no longer an element of $W^{k, p}$ for $p$ greater than some number?

Consider the function $u(x) = x^{\frac{1}{2}}$ on the domain $[0, 1]$. This function is an element of $W^{1, 1}$ and $W^{1, \infty}$ but not $W^{1, 2}$ as for $W^{1, 1}$, we have $\Vert\frac{\...
1
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1answer
41 views

Closed graph theorem; exercise

Let $E$ be a Banach space and let $T:E\to E^{\star}$ be a linear operator satisfying $\langle Tx,x\rangle\geq 0$ $\forall x\in E$. Prove that $T$ is a bounded operator. My Solution (but I have ...
2
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0answers
13 views

Different definitions of Besov norm/space

I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent. Victor I. Burenkov - Sobolev Spaces On Domains. Given $f:\mathbb{R}^n \...
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1answer
16 views

Weak and Weak$^{\star}$ topologies: Annihilator

Exercise: Let $E$ be a Banach space. Let $M\subset E$ be a linear subspace and let $f_0\in E^{\star}$. Prove that there exists some $g_0\in M^{\perp}$ s.t. \begin{equation}\inf_{g\in M^{\perp}}\Vert ...
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0answers
32 views

Functional analysis : show that the inf is attained

I'm a beginner in functional analysis and I'm trying to solve the following problem: $$ \alpha > 0,\;\; H^{\alpha}= \{ u \in \mathbb{R}^\mathbb{N} \ \mathrm{ such }\ \mathrm{that} \ \sum_{n=0}^{...
21
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8answers
2k views

Why is uniqueness important for PDEs?

Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, ...
1
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2answers
40 views

Every bounded linear operator $T$ between real Hilbert spaces is $T(x) = \sum \langle x,f_j\rangle\, e_j$

Let $T:H_1 \rightarrow H_2$, where $H_1$ and $H_2$ are real hilbert spaces and $T$ is a bounded linear operator. Prove the following: suppose $\{e_j\}$ an orthonormal basis for $H_2$, show that ...
2
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1answer
32 views

Convex basis of functions

I'm looking for a set of convex functions which is forms a basis for $C^1(\mathbb{R})$? Most of the basises I know are polynomials or Fourier basis but I was wondering if there was a basis of convex ...
2
votes
1answer
18 views

Is $X^*$ complete with weak*-topology

Suppose $X$ is a topological vector space, $X^*$ is its topological dual space. Let the topology of $X^*$ is weak*-topology, Is $X^*$ complete? Suppose $f_s$ is a Cauchy net in $X^*$, it is easy to ...
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0answers
69 views

Norm of gradient of velocity field

If $\mathbf{u}(x,y,z,t)=(u,v,w):\mathbb{R}^3\times[0,+\infty)\to\mathbb{R}^3$ denotes a velocity field, what is the definition for $\|\nabla\mathbf{u}\|_{L^{\infty}}$? I know that $\nabla\mathbf{u}$ ...
4
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1answer
60 views

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying: i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$. ii) there is some $\theta \in (0,1)$ ...
6
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3answers
70 views

Show that linear functional $L(f) = \int_0^1 f(x) dx$ is continuous

Let $(C[0,1], d_1)$ be a metric space of all continuous functions $f:[0,1] \to \mathbb{R}$, $d_1$ is the $L_1$ metric $$d_1(f,g) = \int\limits_0^1 |f(x) - g(x)| dx$$ Show that linear functional $L(...
1
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1answer
50 views

Show all sequence of $l^1$ with $|x_n|\leq \frac{1}{n^2}$ is compact.

Could you help me to check my proof: let $\{x^k\}$ be a sequence in such set, we use Cantor's diagonal argument to show the existence of convergent subsequence. There exists a subsequence $\{x^{\...
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0answers
10 views

Questions about the regularity of the solution of the heat equation in a bounded domain

I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$ Here is the statement and ...
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3answers
81 views

$\ker ST=\ker T$

Let $S$ and $T$ be linear maps between vector spaces such that the composition $ST$ makes sense. Clearly, $\ker ST\supseteq \ker T$. The two instances that come to my mind for having an equality in ...
1
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1answer
32 views

If $\mathcal H$ is the closure of the set $D$ of divergence-free smooth functions in $L^2$, then $H_0^1∩\mathcal H$ is the closure of $D$ in $H_0^1$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$\mathfrak D:=\left\{u\in\mathcal D:\nabla\cdot u=0\right\}$$ $H:=H_0^1(\Omega,\mathbb R^d)$...
1
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1answer
38 views

What is the Hilbert adjoint operator of this bounded linear operator?

Let $H$ be a Hilbert space, and let $z \in H$. Let $T_z \colon H \to K$, where $K$ is the field of scalars for $H$ and $K$ is either $\mathbb{R}$ or $\mathbb{C}$, be defined by $$ T_z (x) \colon= \...
0
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0answers
54 views

Two inner products in one vector space.

Please, can you help me answer the some following questions? In theory functional analysis. At first, I want to consider finitely dimensional vector space V over field K(real or complex). Now, if it ...
2
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0answers
17 views

Complex interpolation between $H^1$ and $L^1$

We have the complex interpolation $[H^1,L^p]_\theta=L^q$ for $1<p<\infty$ where $H^1$ is the Hardy space and $\frac{1}{q}=1-\theta+\frac{\theta}{p}$. This raises the obvious question as to ...
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1answer
36 views

A basis of $L^2$

I would like to ask you a question that is there a basis of the space $L^2(\Omega,\mathcal{F},\mathbb{P})$, where $\mathbb{P}$-probability measure?
2
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0answers
46 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
2
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1answer
22 views

Using scaling arguments to determine relationships between Sobolev spaces?

I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: ...
14
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2answers
767 views

Bounding $\int_0^1 f(x) dx $ under the condition $\int_0^1 f'(x)^2 dx \le 1$

Any tips on how to solve this? Problem 1.1.28 (Fa87) Let $S$ be the set of all real $C^1$ functions $f$ on $[0, 1]$ such that $f(0) = 0$ and $$\int_0^1 f'(x)^2 dx \le 1 \;. $$ Define ...
-1
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0answers
13 views

What is the difference between sequence space of bounded complex sequences compared to sequence spaces of bounded and unbounded complex sequences?

Both spaces have different measures as folllows (click to see) Bounded sequence space measure and Bounded and unbounded sequence space measure
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1answer
25 views

Tensor products $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$

Can anybody enlighten me, where the tensor products of the spaces of summable functions $L_1(\mu)\widetilde{\otimes}_{\pi} L_1(\nu)$ and $L_1(\mu)\widetilde{\otimes}_{\varepsilon} L_1(\nu)$ are ...
2
votes
0answers
40 views

Finding closure of image of operator

I'm working on an old exam problem: Define for $u \in C^2([-1,1])$ the operator $L$ by $[Lu](x) = - \frac{d}{dx} \left( (1-x^2) u'(x) \right)$. Set $\Omega = \{ Lu \mid u \in C^2([-1,1]) \}$. Find the ...
0
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0answers
12 views

Extension of semi-norm in locally convex topological space

Suppose $X$ is a locally convex topological space,$M$ is a subspace. Suppose $p$ is a continuous semi-norm on $M$. Is it possible to extend $p$ to be a continuous semi-norm on $X$? What if $X$ is not ...
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2answers
38 views

Uniform closure of densely continuous functions

Consider the collection of those $\mathbb{R}$-valued functions on an interval $I\subseteq\mathbb{R}$, which have a dense set of points of continuity. I would expect this collection to be closed under ...
2
votes
2answers
38 views

Weak problem formulation for PDE and boundary conditions

Consider the following example: $$ - \Delta u = f \mbox{ in } \Omega, $$ $$ u = 0 \mbox{ on } \Gamma, $$ Here $\Gamma$ is boundary of $\Omega$. To produce weak formulation we multiply by arbitrary $v$ ...
3
votes
2answers
47 views

L^1 convergence and limsup of convergent sequence

I have to solve this exercise: let $f_n$ be a sequence of positive real function defined on a measure space $(X,M,\mu)$ such that $f_n\in L^1(\mu)$ $\forall n\in \mathbb{N}$ and $f_n$ is convergent in ...
7
votes
0answers
51 views

How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}udy. $$ But for the mean-value property of the Heat equation, ...
1
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1answer
65 views

Continuous semi-norms on subspace

Suppose $X$ is a locally convex topological vector space, let $P$ be the set of all continuous semi-norms on $X$. Suppose $M$ is a subspace of $X$, denote $P|_M$ as the set of semi-norms in $P$ ...
2
votes
1answer
26 views

SOT-isomorphic C*-algebras

Suppose that $A, B \subset B(\mathcal{H})$ are $C^*$-algebras. Assume that $\{p_n\} \subset B(\mathcal{H})$ is a monotone sequence of projections such that: $p_n \rightarrow 1$ in strong operator ...
4
votes
0answers
28 views

Proof validation: complete set, change of variable

Let $\phi(x) \in \mathcal{C}^1([0,1])$ be a real valued function such that: $$\begin{cases} \phi'(x) > 0 & \forall x \in [0,1] \\ \phi(0) = 0 \\ \phi(1) = 1. \end{cases}$$ I'm asked to prove ...
0
votes
1answer
54 views

Do we have $\|x\|_p \le \|x\|_q$, where $x$ is bounded and $p>q$? [closed]

Assume that $x\in L_p(0,\infty) \cap L_q(0,\infty)$, where $p>q$. Do we have $$\|x\|_p \le \|x\|_q?$$ where $x$ is bounded.
2
votes
1answer
37 views

Construct a sequence in Banach Space

Prove the equivalence between: $\forall x \in B_E = \{y \in E:\|y\| \leq 1 \}$ $\exists (x_n) \subset E$ such that $\|x_n\|=1$ and $x_n \rightarrow_w x$ (weak convergence). $\exists (x_n) \subset E$ ...
2
votes
1answer
33 views

Closure of an operator

I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$ $$A_0= f^{(4)}-f^{(6)}$$ $$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(...
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votes
3answers
44 views

Value of $f^2(4)+g^2(4)$ [duplicate]

If $f(x)=g'(x),g(x)=-f'(x)$ for all real $x$ and $f(2)=4=f'(2)$ then value of $f^2(4)+g^2(4)$ is ? Now the above is true when we have a constant function with constant $0$. But then that would not ...
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0answers
25 views

Problem of a functional on Orlicz spaces

Please i have a problem with the lemma 3.1 in the following paper : https://www.researchgate.net/publication/277715258 Please how to use the conditions to find that $g(t)>0$ for $t$ ...
4
votes
1answer
105 views

A Continuous Function with a Divergent Fourier Series

This is a Q&A; I hope simply posting a question and then answering it is the right protocol. This is stuff I thought everybody knew, but in at least two recent threads it's turned out to be somewhat ...
1
vote
1answer
34 views

$L:L^2([0,1])\rightarrow L^2([0,1])$, $Lf(x)=\frac1{x+1}f(x)$ not compact

Let $L:L^2([0,1])\rightarrow L^2([0,1])$ given by $Lf(x)=\frac1{x+1}f(x)$. I want to show that $L$ is continuous but not compact. The boundness follows by $\|Lf\|_2\leq \|f\|_2 $ since $\frac1{x+...
0
votes
0answers
23 views

For $u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$, is $[\partial_t (u_h(t))]_{x_j} = \partial_t[(u_{x_j})_h(t)]?$

Let $u$ belong to $L^2(0,T;H^1(\Omega))$ and $u_t \in L^2(0,T;(H^1(\Omega))^*)$ and define the function $$u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$$ Is it true that $$[\partial_t (u_h(t))]_{...
0
votes
0answers
33 views

Lesbesgue spaces involving time

If I have a function which lives in $f(x,t)\in L^2(0,T; H^{\frac{1}{2}})\cap L^{\infty}(0,T; L^2)$ for a certain time interval. I also know that $\partial_t \ f(x,t)\in L^{2}(0,T;H^{-1})$. Can I ...
2
votes
0answers
26 views

A Banach space $X$ is reflexive iff $X^*$ is reflexive [duplicate]

I have already shown that if $X$ is reflexive then $X^*$ is reflexive, but I need some help on the other direction. The canonical mapping is defined by $$ J : X \to X^{**}, \ J(x) (f) = f(x)$$ For ...
1
vote
1answer
25 views

If $U$ is a vector subspace of a Hilbert space $H$, then each $x∈H$ acts on $U$ as a bounded linear function $〈x〉$. Is $x↦〈x〉$ injective?

If $H$ is a $\mathbb R$-Hilbert space, then the duality pairing $$\langle\;\cdot\;,\;\cdot\;\rangle_{H,\:H'}:H\times H'\;,\;\;\;(x,\Phi)\mapsto\Phi(x)$$ can be considered as being a mapping $H\times H\...
2
votes
0answers
63 views

Misunderstanding of Atiyah-Singer

I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies Analytical index = ...
0
votes
0answers
12 views

Diagonal process for subsequences

I am reading Peter Lax Functional analysis and I do not understood how he shows that exist a sequence of $z_n$ such that for every j $lim_n m_j(z_n) $ exists.
1
vote
0answers
38 views

Relationship between the distributional Laplacian and the weak Laplacian

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the $L^2(\Omega)$- or $L^2(\Omega,\mathbb R^d)$-inner product (depending on the context) $\mathcal ...
1
vote
1answer
63 views

Topology for Hardy spaces

Let $\Omega\subset \mathbb{C}$ be an open set (of the complex plane) and let $\mathcal{H}(\Omega)$ be the algebra of analytic functions on $\Omega$ endowed with the topology of compact convergence (...