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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Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } ...
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2answers
252 views

Isometry on a dense sub-space of a Banach space?

Let $X$ be a Banach space and let $D$ be a dense sub-space of $X$. I don't know if the following fact is true: Fact: For every (linear) isometry $T\in\operatorname{Iso}(X)$ and for every ...
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1answer
263 views

Showing a sequence of integrals converges to zero

Let $\varphi$ be a complex-valued function which is analytic on $\{z \in \mathbb C : |z| \leq 2\}$, let $\gamma$ be the unit circle in the complex plane, and define $$ F_n(z) = \int_\gamma ...
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1answer
592 views

Closed unit ball is compact?

Let $X$ be a Banach space and let $\operatorname{Lip}_{0}(X)$ be the space of all real-valued Lipschitz functions which vanish at $0$. The space $\operatorname{Lip}_{0}(X)$ is a Banach space when it ...
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1answer
220 views

Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
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198 views

A type of local minimum (2)

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is a Lipschitz graph. $S \subset \partial \Omega$ is measurable and $H^{n-1}(S)>0$. ...
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1answer
412 views

Spectrum of composition of convolution and multiplication

Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$ defined as composition of convolution and multiplication, $Tf := (af) * g$ where $g$ is in $L^2$ and $a$ is a bounded function. ...
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236 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
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453 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
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568 views

Compact set of probability measures

I think I can solve the following exercise if X is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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936 views

Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
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1k views

The definition of metric space,topological space

I have read some books in analysis,all of them define metric space,topological space or vector space directly,without any reason. Therefore, I want to know the background of the definition, the ...
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1k views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
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1answer
358 views

How to prove Halmos’s Inequality

How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from ...
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2k views

Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$

I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrasingly weak - I would like to show that, for a real number $p ...
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1k views

$C_c(X)$ dense in $L_1(X)$

I did the following homework, can you tell me if my answer is correct? Prove that for any $\sigma$-compact, locally compact Hausdorff space $\Omega$ equipped with a Radon measure $\mu$ the set of ...
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361 views

Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
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380 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
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1answer
785 views

Hahn-Banach theorem: 2 versions

I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as: Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of ...
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1answer
1k views

Metric and Topological structures induced by a norm

While proving that some normed spaces were complete, two questions came to my mind. They relate the topological and the metric structures induced by a norm. 1) Is it possible to find two equivalent ...
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901 views

Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?

In Lieb and Loss's Analysis, I saw that they mentioned $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the ...
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Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
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772 views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
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3answers
351 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
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1answer
1k views

Equicontinuity on a compact metric space turns pointwise to uniform convergence

I know that If $\{f_n\}$ is an equicontinuous sequence, defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly. I'm having trouble ...
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1answer
586 views

Are a weak derivatives and distributional derivatives are different?

For simplicity, given a real function $f\in L^1_{loc}(\Omega)$, we define both weak or distributional derivatives by $\int f'\phi = - \int f \phi'$ for all test functions $\phi$. Now, take $\Omega = ...
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1answer
377 views

Trace class for operators

Let $ \mathcal{H} $ be a Hilbert space and $ T: \mathcal{H} \to \mathcal{H} $ a bounded linear operator. The $ n $-th singular number $ {\mu_{n}}(T) $ of $ T $ is defined as the distance from $ T $ ...
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1answer
88 views

Self adjoints and unitaries in a banach * algebra

Are the spectra of self adjoints and unitaries in banach * algebras necessarily a subset of the reals and the unit circle respectively? The proofs I know for C* algebras use the continuous functional ...
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1answer
356 views

A paradox on Hilbert spaces and their duals

I am making some elementary mistakes here. Could you please help me point out the problems? Thank you very much! Suppose on some space $H$ we have two inner products, which make $H$ after completion ...
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1answer
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Properties of dual spaces of sequence spaces

Can you tell me if I got the following homework right? Nitpicking is welcome. a) Recall that $$ c_0 (\mathbb{N}) = \{ f: \mathbb{N} \rightarrow \mathbb{C} \mid \lim_{n \rightarrow \infty } f(n) ...
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On Some Properties of Hölder Continuous Functions

The function space $H^{\alpha} (\Omega)$ for $0 < \alpha \le 1$, is the set of functions: $$\{ f \in C^0(\Omega) : \sup_{x \neq y} \dfrac{|f(x) - f(y)|}{|x-y|^{\alpha}} < \infty \}$$ with the ...
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1answer
959 views

Dual space of $H^1$

It holds that $W^{1,2}=H^1 \subset L^2 \subset H^{-1}$. This is clear since for every $v \in H^1(U)$ $u \rightarrow (u,v)_{H^1}$ is an element of $H^{-1}$. Moreover for every $v \in L^2(U)$ $u ...
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1answer
2k views

Norm on a Hölder's space

I want to prove that Hölder space is a Banach space under the "Hölder Norm" ie. $\|\cdot\|_{C^{k,\alpha}}$. Any hints would be appreciable .
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2answers
169 views

General “Hodge theorem”

I know basically zero Hodge theory, so this question might be weird. Let $$A \stackrel{S}{\longrightarrow} B \stackrel{T}{\longrightarrow} C$$ be a sequence of closed, densely defined maps of ...
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1answer
388 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
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1answer
381 views

Does the Gelfand transformation on $\ell^1(\mathbb Z)$ possess a continuous inverse on its image?

I am interested in the Gelfand transformation $$ \Phi\colon\ell^1(\mathbb Z)\to\mathcal C(\mathbb T),\quad a\mapsto\sum_{n\in\mathbb Z}a_n z^n. $$ This is an injective homomorphism of Banach algebras. ...
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2answers
234 views

Do we need to identify dual spaces in PDEs?

In PDEs we often use the fact that we can identify dual spaces eg. $L^2(0,T;V)^* = L^2(0,T;V^*)$ in the sense that $$u_t + Au = f$$ where $u_t$, $f \in L^2(0,T;V^*)$ and $A:L^2(0,T;V) \to ...
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1answer
333 views

Dual of “Dual of Fréchet Space with Weak*-Topology” Equals Dual of “Dual of Fréchet Space with Topology of Compact Convergence”

Let $X$ be a Fréchet space. I know that: Closed convex, balanced, hulls of compact subsets of $X$ are compact. Let ${X^*}$ denote the (topological) dual space. I know that: ${X^*}$ is also the ...
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1answer
196 views

Is the identity map $id: H^2(-\pi,\pi) \to L^2(-\pi,\pi)$ Hilbert-Schmidt?

Let $H_1, H_2$ be Hilbert spaces. A linear operator $A: H_1 \to H_2$ is Hilbert-Schmidt iff for some orthonormal basis $\lbrace e_n : ~ n \in \mathbb{N} \rbrace$ of $H_1$ the sum $\sum_{n \in ...
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1answer
326 views

An application of J.-L. Lion's Lemma

Let $X,Y$ and $Z$ be three Banach spaces with norms $\|\cdot\|_X$, $\|\cdot\|_Y$,$\|\cdot\|_Z$. Assume that $X\subset Y$ with compact "injection" and that $Y\subset Z$ with continuous injection. Then ...
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916 views

Completeness of a finite direct sum of closed subspaces of $L^2$

Let $X_1$ and $X_2$ be real-valued square-integrable random variables defined on a probability space $(\Omega, {\cal F},P)$. For $i=1,2$, set $$ A_i := \{g(X_i)\in L^2 \mid g \text{ is some Borel ...
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143 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
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398 views

Definition of Banach limit

In my Bachelor Thesis I have defined a Banach limit as a functional $LIM: l^\infty (\mathbb{N})\rightarrow \mathbb R$ that has the following properties: B1 If $(x_n)$ is a convergent sequence, then ...
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1answer
216 views

supremum of $\int |\int f(x)-f(y)\,dy| \,dx $.

Let $A_k$, $k\in \mathbb{N}$, be the family of $C^\infty([0,1])$ functions defined by $$ A_k=\{||f^{(j)}||_{\infty}\le 1,\;\;0\le j \le k \} $$ where $||\cdot||_{\infty}$ denotes the supremum norm ...
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1answer
739 views

Mazur's Lemma (can't find a proof anywhere)

If we have a normed vector space $X$, and $x_n \to x$ weakly, then it's clear how to show there's a sequence $y_n \to x$ strongly (since the weak closure and strong closure of a convex set coincide, ...
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1answer
812 views

The dual of a Fréchet space.

Let $\mathcal{F}$ be a Fréchet space (locally convex, Hausdorff, metrizable, with a family of seminorms ${\|~\|_n}$). I've read that the dual $\mathcal{F}^*$ is never a Fréchet space, unless ...
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1answer
450 views

On Pitt's theorem

The famous Pitt's theorem asserts that if $p>q$ then each bounded operator $T\colon \ell^p\to\ell^q$ is compact. Since $\ell^p$ and $\ell^q$ are incomparable ($p\neq q$, $p,q\geq 1$), each operator ...
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1answer
96 views

Is the complex Banach space $C([0,1])$ dual to any Banach Space?

I've been able to show that the extreme points of $C([0,1])$ are the continuous functions that take values on the unit circle. However, I'm not sure how to reason from here as to whether or not it is ...
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1answer
129 views

The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
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1answer
435 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...