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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Bochner: Absolute Integrability

For a Bochner measurable function it holds: $$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$ for any positive measure $\lambda\geq 0$. The one direction is relatively simple when ...
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1answer
47 views

Integral operator on $L^p$ is compact

Let $(X,\Omega,\mu)$ be an arbitrary measure space, $1<p<\infty$ , and $\frac{1}{p}+ \frac{1}{q} = 1$. If $k:X. X\to \Bbb C$ is an $\Omega.\Omega-$ measurable function such that $$M = [\int ...
2
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1answer
18 views

An abstract a priori estimate in finite element method

Let $V$ and $K$ be Banach spaces (with norms $\|\cdot\|_V$ and $\|\cdot\|_K$ resp.) and suppose that there is a compact linear embedding $K\hookrightarrow V$. Furthermore, let $P_n$ be a family of ...
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0answers
12 views

Volterra operator and completely continuous operators

Consider the Volterra operator $V$ defined here. Let me give some definitions first: [Dunford-Pettis] We say that a bounded linear operator $D:L_1[0,1]\to L_1[0,1]$ is Dunford-Pettis if it sends ...
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0answers
9 views

Is Sobolev regularity propagated under evolution?

Given a well-posed initial problem in a domain $\Omega$ of the form: \begin{equation} \square\phi=f \end{equation} where $\square$ is the D'Alamebrt operator, $f\in L^{2}(\Omega)$, with initial ...
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0answers
15 views

Prove the property of the theorem

Before you submit theorem for which I am concerned will first give some necessary data $1)$ Proposition 1: Let be $X$ a normed space, $Y$ unitary space and $A:X\rightarrow Y$ linear operator. The ...
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1answer
15 views

subadditivity and continuity at zero implies continuity at all points

If $p$ is a subadditive functional on a normed space $X$ and is continious at $0$ and $p(0)=0$. To show $p$ is continious for all $x \in X$. This is a problem from Kreyszig's Introductory Functional ...
3
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1answer
31 views

If $T$ is self-adjoint, is the set of power series in $T$ closed?

If $T$ is a bounded self-adjoint operator on a Hilbert space, is the set of convergent power series in $T$ closed in the norm topology? I ask because I'm reading some spectral theorems and I was ...
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0answers
18 views

integration formula

help me please all these functions are regular. How we can found this formulation $$ \displaystyle\int_{\Omega} (f(u)-f(k)) \nabla p(g(u)-g(k)) \xi dx = - \displaystyle\int_{\Omega} H(u,k) \nabla \xi ...
5
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1answer
63 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
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1answer
49 views

Misunderstanding about Laplace operator

Let $\Omega$ be a bounded subset of $\mathbb{R}^n$. We know that the Laplace operator \begin{align} \Delta \colon H_0^1(\Omega) \to L^2(\Omega) \end{align} admits an inverse operator \begin{align} A ...
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1answer
17 views

Extending Banach Space of Functions

The idea is that one could in principle consider the space of functions: $$\{f:\Omega\to V\}$$ with pointwise operations and uniform convergence: $$f_\lambda\to f:\iff\|f_\lambda-f\|_\infty\to 0$$ ...
2
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0answers
29 views

Dual subfactor and commutant

Let $(N \subset M)$ be a subfactor and $N \subset M \subset M_1$ the basic construction. Question: Is $(M \subset M_1) \simeq (M' \subset N')$? Else in which generic case it's true? What's the ...
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14 views

numerical+variational mixed optimization $\min_{l,u,f(.)} (u-l)+\int_{-1}^1 (f'(x)-x)^2dx$

Given a function $g(x)$ and its domain, we want to get another function $f(x)$ whose derivative approximate $g(x)$ well, but so that $f(x)$ itself has small variation.for example, for ...
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0answers
27 views

Is every subspace of a normed linear space which is not closed a hyperspace.

Let $B \subset X$ where $X$ is a normed linear space over $\mathbb{R}$ and $B$ is a proper subspace. If $B$ is not closed, is $B$ necessarily a hyperspace(maximal proper subspace) in $X$. I attempted ...
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1answer
34 views

Periodic Laplace operator non closed in $ C^2(0,L)$

How can I show that the Laplacian operator is not closed in the domain $D=\{f \in C^2(0,L) \mid \mbox{ f is vanishing in a neighborhood of 0 and L } \}$ for a fixed $L$? And how can I show that it is ...
2
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2answers
78 views

Volterra operator is completely continuous

Let $\mu$ be the Lebesgue measure on $[0,1]$ on the borelians, and consider the Volterra operator $V:L^1[0,1]\to C[0,1]$ given by $$ Vf(t)=\int_0^t f d \mu $$ So, I want to show the following ...
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1answer
14 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
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4answers
538 views

An idempotent operator is compact if and only if it is of finite rank

Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank.
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2answers
41 views

It's the discrete topology.

I have to proof that if I have $(X,\tau)$ and $(Y,\delta)$ two topological spaces, if every function $f:X\longrightarrow Y$ is continuos then $\tau$ is the discrete topology. I don't know what is the ...
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0answers
18 views

Radon-Nikodym: Complex Measures

Let $\Omega$ be a measureble space and $\mu$ a complex measure. (Note that this implies that the measure is finite.) Consider an absolutely continuous complex measure $\nu\ll\mu$. Then: $$\nu=\int ...
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61 views
+50

Isomorphism between $C^\infty_0(B_1)$ and $\mathscr{S}(\mathbb{R}^n)$

Background: Related question I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in ...
2
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0answers
33 views

Riesz-Markov-Kakutani Theorem: Various Versions

The Riesz-Markov-Kakutani theorem usually comes in various versions. So I'm a little bit confused and wondering which of these are right. Let $\Omega$ be a locally compact space. Then: Complex ...
2
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1answer
30 views

Markov processes on function spaces

Is there any reference on Continuous time Markov process whose state space is infinite dimensional function spaces, such as the space of continuous functions $C(R^d)$? It seems Dirichlet Form is a ...
2
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1answer
35 views

Three-space property

I have found two definitions of a three-space property. One definition is: $(P)$ is a three-space property if whenever $E$ Banach space, $F\subseteq E$ is a closed linear subspace and two of the ...
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1answer
21 views

Question about proving a function on linear space to be norm

Let $X$ denote the linear space of all polynomials in one variable with coefficients in $\mathbb{K}$. (Where $\mathbb{K}$ denotes $\mathbb{C}$ or $\mathbb{R}$). For $p \in X$ with $p(t) = a_0 + a_1t ...
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1answer
44 views

Is a functional derivative a generalized function?

I am just now learning about elementary distribution theory, and it seems that theory may bear on the topic of functional differentiation, which I've encountered in some books on quantum field theory ...
1
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1answer
39 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions: $$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$ where $f_n(x)= ...
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0answers
13 views

Solution set of linear operator equations

Suppose $\mathcal{X}$ and $\mathcal{Y}$ are two Hilbert spaces. Let $A:\mathcal{X} \mapsto \mathcal{Y}$ be a bounded linear operator. Consider a linear operator equation $Ax=b$. My question is what ...
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32 views

when does classical theory of parabolic PDE fail

I am reading a paper on degenerate parabolic PDE and I am confused about the following statement. " Diffusion coefficient is $mu^{m-1}$, and it vanishes when $u=0$. Hence at all those points where ...
2
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1answer
19 views

$\sup$ norm of a function

The following is an example of Murphy's C*-algebras and operator theory: I do not know how he concludes $$\int_0^1 |k(s,t) - k(s',t)||f(t)| dt \leq \sup|k(s,t) - k(s',t)|||f||_\infty$$ Please help ...
2
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0answers
30 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
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0answers
26 views

Motivation for the notion of locally convex topological vector space

Is the only motivation for the notion of locally convex topological vector space that the local bases have some nice property i.e. convex, balanced, absorbing ?
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1answer
98 views

What is the closure of the Laplacian on $L^2(0, \infty)$ with domain $D(\Delta):=C^\infty_0(0, \infty)$?

Let $\Delta$ be the operator on $L^2(0, \infty)$ defined as follows: $\Delta \phi:= \phi''$, with domain $D(\Delta):=C^\infty_0(0, \infty)$. Is $\Delta$ closed or closable? In the case, what is its ...
4
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1answer
44 views

Is there $u,v\in L(E): uv-vu=id_E$

Let $E$ be a normed vector space over $\mathbb{R}$. Is there continuous linear transformations $u$ and $v$ such that: $$uv-vu=id_E$$ (.ie $\forall x\in E:u(v(x))-v(u(x))=x$) I suspect that the ...
1
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1answer
23 views

Intersection of nested closed balls in a normed linear space [on hold]

Let $B_{1}\supset B_{2} \supset ...$ be closed balls in a normed space X, where $B_{n}=B(x_{n},r_{n})=\{x:\|x-x_{n}\|\leq r_{n}\}$, $r_{n}\geq r > 0$. Is $\bigcap_{n=1}^{\infty} B_{n}$ nonempty? Is ...
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13 views

Intersection of closed bounded subsets in a Banach space [on hold]

Prove or disprove : In a Banach space every nested sequence of closed bounded subsets has a non-empty intersection.
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32 views

to prove that a metric space which is not complete [on hold]

Given $C_{\infty}= \{x=(x_n) :x_n \in \mathbb{R}\ \text{and}\  \exists\ n(x)\in \mathbb{N} $ s.t $x_n = 0\ \forall\ n > n(x)\}$  Where each element is sequence of type ...
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0answers
26 views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
2
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2answers
47 views

Frechet differentiable implies reflexive?

Note: The question has been cross-posted (and answered) on MathOverflow here. Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive?
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+100

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
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3answers
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Matrices with $n$ eigenvalues [on hold]

My question is: how can I prove that the set of matrices with $n$ distinct eigenvalues is open in the space of $n\times n$ matrices over $\mathbb{C}$ ? using a perturbative argument.
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0answers
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Given a block-form contraction operator $X$, can we write $I-XX^*$ as $PP^*$ with a nice block form of $P$?

Suppose the operator $$X = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$$ is contractive, where $A, B, C$ and $D$ are themselves bounded operators, then we know that $I - XX^* = PP^*$ for ...
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0answers
9 views

Does symmetric decreasing rearrangement of a smooth function preserves smoothness?

Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of ...
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1answer
31 views

Intuition $C^*$-identity

Let X be a Banach algebra and $x,y\in X$ by definition then $\Vert xy\Vert\leq\Vert x\Vert \Vert y\Vert$ and the intuition could be that it makes multiplication continuous, which is a nice property. ...
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0answers
9 views

The regularity of Dirichlet form in Besov space

Let $C_{0}(\mathbb{R}^n)$ denote the set of continuous functions in $\mathbb{R}^n$ with compact support, $C_0^\infty(\mathbb{R}^n)$ denote the set of infinite differentiable functions in ...
2
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1answer
33 views

Fubini's theorem for complete $\sigma$-algebras vs. non-complete $\sigma$-algebras

Suppose $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are both complete measure spaces. Consider the following two measure spaces: $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ and $(X ...
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1answer
13 views

Most general type of $L^p(X,\ V)$ space where compactly-supported continuous functions are dense

Let $(X,\ \tau)$ be a topological (locally compact?) space, $(X,\ \mathcal{F},\ \mu)$ a measure space, $(V,\ \|\cdot\|)$ a Banach space, $1\leq p < \infty$ and $\|\cdot\|_p$ a function defined for ...
3
votes
2answers
250 views

Heat equation and semigroup theory.

Theorem: Let $X$ be a Banach space, $\{T(t)\}_{t\geq 0}$ a $C_0$-semigroup on $X$ and $U_0\in X$. If $A:D(A)\subset X\to X$ is the infinitesimal generator of $\{T(t)\}_{t\geq0}$, then the function ...
4
votes
1answer
201 views

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...