1
vote
1answer
33 views

Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
0
votes
1answer
23 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 ...
2
votes
1answer
47 views

Positive Linear Transformations: What good for?

Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct ...
3
votes
1answer
38 views

What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
1
vote
1answer
60 views

Definition of a Bounded Operator and Some Intuition on the Definition of the Norm

I am confused about the definition of a bounded operator (which is probably a consequence of my unsatisfactory understanding of bounedeness and local boundedness). The definition is ...
2
votes
3answers
121 views

Compact subsets of function spaces, geometry

The subset is called compact when every open cover contains a finite subcover. In Euclidean spaces, it is easy to visualize this by imagining some open ball that contains this set, thinking about the ...
4
votes
1answer
98 views

Geometric intuition behind the Uniform Boundedness Principle

Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
6
votes
2answers
167 views

Discreteness of eigenvalues for certain operators - can this approach be made rigorous?

I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
3
votes
1answer
112 views

What is the use of $H_s$ for non-integer $s$?

So we have the whole set of theory for Sobolev spaces \begin{equation} H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\}, \end{equation} and we know that ...
2
votes
1answer
56 views

So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?

For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation} H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\}, \end{equation} where $\mathcal{S}'$ is the space of ...
0
votes
1answer
94 views

How does a myopic interpret Wiener's Tauberian?

I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made ...
2
votes
0answers
116 views

Intuition behind the proof for Wiener's theorem?

I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind ...
9
votes
5answers
977 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 ...
7
votes
4answers
748 views

Is there any motivation for Zorn's Lemma?

I have been reading Kreyszig's book on functional analysis, where it uses Zorn's lemma to prove the Hahn Banach theorem. However I don't quite get what Zorn's lemma is saying. I understand that it ...
3
votes
1answer
386 views

Intuition behind Sobolev norm

This morning I was thinking at the following (simple) fact. Let us consider $[0, 1] \to \mathbb{R}$ functions and define a linear functional $$F(u)=u(1)-u(0).$$ $F$ is not continuous on $L^2(0, 1)$ ...
1
vote
1answer
654 views

Seminorms and norms

Suppose we have the following lemma: Lemma If $E_0 \hookrightarrow E$, and $E_0$ is a closed subspace then $E/E_0$ is a normed space and for $[x] \in E/E_0$ its norm is given by $||[x]|| = ...
3
votes
3answers
282 views

Intuition Behind Balanced Sets

Suppose $B \subset X$ where $X$ is a vector space. $B$ is called balanced if $\alpha B \subset B$ for every $\alpha \in \Phi$ with $|\alpha| \leq 1$. Note that $\Phi = \textbf{R}$ or $\Phi = ...