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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Applications of Banach Algebras and Operator Algebras

I am trying to learn operator algebra theory (I am tempted to start with Douglas' "Banach Algebra Techniques in Operator Theory"). One aspect that I am curious about is whether there are significant ...
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525 views

Is the natural map $L^p(X) \otimes L^p(Y) \to L^p(X \times Y)$ injective?

Let $X,Y$ be $\sigma$-finite measure spaces, and let $L^p(X) \otimes L^p(Y)$ be the algebraic tensor product. The product has a natural map into $L^p(X \times Y)$ which takes $\sum a_{ij} f_i \otimes ...
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198 views

Prove that there exist linear functionals $L_1, L_2$ on $X$

Let $X$ be a linear space, $p, q$ sublinear functionals on $X$, and $L$ a linear functional on $X$ such that $|L(x)| ≤ p(x) + q(x),$ for all $x ∈ X$. Prove that there exist linear functionals $L_1, ...
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383 views

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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253 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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264 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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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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430 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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110 views

If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But ...
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241 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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462 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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587 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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962 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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8answers
571 views

For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$

Suppose $f:[0,1]\to \mathbb{R}$ is continuous. Show that  $$\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$$ My answer so far: First I want to assume that $f\in C^1$. Then  $$n\int_0^1f(x)x^n\,dx ...
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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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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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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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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 $p ...
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$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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366 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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494 views

Counterexample for the solvability of $-\Delta u = f$ for $f\in C^2$

I'm learning for an exam and I'm surprised by the following statement that is given without proof or example: Let $\Omega\subset\mathbb{R}^n$ be open, bounded and connected and let $f\in ...
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2k views

Invertibility of compact operators in infinite-dimensional Banach spaces

Let $X$ be an infinite-dimensional Banach space, and $T$ a compact operator from $X$ to $X$. Why must $0$ then be a spectral value for $T$? I believe this is equivalent to saying that $T$ is not ...
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1answer
384 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
797 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
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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
931 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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780 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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1answer
597 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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397 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
89 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
362 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
2k views

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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4answers
100 views

What's so special about $p=2$ for the $L^p$ spaces?

The Banach space dual of $L^p$ is $L^q$, where $q=\frac{p}{p-1}$, but I don't really understand the motivation behind this. In particular, I find it kind of surprising that the only $L^p$ space whose ...
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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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695 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
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171 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
393 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
385 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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239 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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349 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
203 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
333 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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2answers
938 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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1answer
70 views

Are distance functions $ d(a_1,x), …, d(a_n,x) $ for an arbitrary metric space linearly independent?

If we have a metric space $ (X,d) $, a finite set of distinct points $ a_1, ..., a_n $, do the distance functions $ d(a_1,x), ..., d(a_n,x) $ have to be linearly independent? That is, if $ ...
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2answers
144 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 ...
9
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2answers
412 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 ...