Tagged Questions

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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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 ...
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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. \inf_{g\in M^{\perp}}\Vert ...
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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 ...
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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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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?
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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 ...
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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: ...
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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 ...
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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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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 ...
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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 ...
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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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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 ...
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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 $v$ ...
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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 ...
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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, ...
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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 $P$ ...
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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 ...
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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 ...
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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.
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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). $\exists (x_n) \subset E$ ...
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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 ...
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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 ...
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 $H\times H\... 0answers 63 views Misunderstanding of Atiyah-Singer I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies Analytical index = ... 0answers 12 views Diagonal process for subsequences I am reading Peter Lax Functional analysis and I do not understood how he shows that exist a sequence of$z_n$such that for every j$lim_n m_j(z_n) $exists. 0answers 38 views Relationship between the distributional Laplacian and the weak Laplacian Let$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 ...
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 (...