1
vote
3answers
38 views

Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
1
vote
1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
1
vote
0answers
77 views

Ascoli-Arzela Theorem [duplicate]

Wikipedia gives the Ascoli-Arzela theorem for a Compact Hausdorff Space. Following through the proof sketch at the end of the linked section, I could not find where the Hausdorff property is used. The ...
1
vote
0answers
24 views

Almost Everywhere Function Space

Problem Let $\Omega$ be a measure space with measure $\mu$ and $V$ a topological vector space not necessarily Hausdorff as well as the function space $\mathcal{F}:=\{f:\Omega\to V\}$ topologized by ...
0
votes
1answer
44 views

Orthogonality on Banach spaces

I got a doubt with a proof in Brezis' Functional Analysis, theorem 2.16. It says Theorem 2.16: Let $G,L \subset E$ be two closed subspaces in a Banach space $E$. Then the following properties are ...
2
votes
0answers
37 views

explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
0
votes
2answers
60 views

Important applications of the Uniform Boundedness Principle

There's like three applications of the uniform boundedness principle in wikipedia: 1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, ...
0
votes
1answer
95 views

Irrational Rotation

Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent; (1) O(z) is dense in S for some z in S, (2) O(z) is dense in $S^1$ for every z in $S^1$, (3) $\sigma$ is ...
6
votes
1answer
96 views

best intuitive books/video lectures to read topology and functional analysis

What are the best intuitive books/video lectures to read topology and functional analysis ? I am aware of basic linear algebra, analysis and measure theory.
0
votes
1answer
40 views

Topology of Normed Space

$(X, \lVert \cdot \rVert)$ is a normed space. Let $x \in X \setminus \{0\}$ and $Y \subset X$ is a subspace. Prove that if $Y$ is open then $Y=X$. Which technique is more useful? We know ...
1
vote
1answer
41 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
2
votes
1answer
65 views

Why the space of probability measures is a subset of the measure space

Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R ...
0
votes
1answer
28 views

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable. It seems very obvious intuitive, but how to write a good solid proof? Notice I take the closure of the span (the ...
0
votes
0answers
23 views

Need help with the proof of the KKM-lemma.

I have been working on the proof of the KKM-lemma, which states Let $\lbrace A_0,A_1,...,A_n \rbrace$ be a closed covering of an $n$-simplex $\sigma=[x_0,...,x_n]$ such that for each face ...
0
votes
1answer
53 views

Balanced Core: $U\text{ open }\implies U^*\text{ open}$

I need one last lemma for the proof of finite dimensional subspaces are closed: Is it true that if a subset is open so is its balanced core??
5
votes
0answers
100 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
0
votes
1answer
38 views

Topological Vector Space: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ...
0
votes
0answers
22 views

reverse-reverse of Michael selection theorem

Let $X\subseteq\mathbb R^d$ be a compact and $Y=\mathbb R^d.$ Let $\Gamma:X\twoheadrightarrow Y$ be a multi-valued map with closed values. Assume that $\Gamma$ admits a continuous (single-valued) ...
2
votes
1answer
27 views

Continuity in the Strong vs Operator topologies on a compact space

Suppose $X$ is a compact space, $H$ is a Hilbert space, and $f:X \rightarrow B(H)$ is continuous when $B(H)$ is given the strong topology. Does this imply that $f$ is continuous when $B(H)$ is given ...
3
votes
1answer
61 views

Find vector space $X$; so that vector space operations are not continuous

How to choose $X$ to be a complex vector space with a topology $\tau$ on $X$; so that vector space operations are not continuous with respect to $\tau$; that is, the mappings, $X\times X \to X: ...
2
votes
1answer
45 views

Prove that scalar multiplication is continuous

Let $X$ be a normed space over scalar field $\mathbb{K}$. I have to show that scalar multiplication map is continuous. So take $X\supset(x_n)\rightarrow x$ and $\mathbb{K}\supset(k_n)\rightarrow k$. ...
2
votes
1answer
49 views

Linear Functional: Continuous? [duplicate]

Given a Banach space: $E$ and chosen a Hamel basis: $\mathcal{B}$ Any vector induces a (noncanonical) algebraic linear functional by: $$\delta:E\to E^*:\delta_b(b'):=\delta_{b,b'}\text{ defined ...
1
vote
1answer
33 views

$\sup_t |T(t)|<+\infty$ implies $\sup_t |T(t)^*|<+\infty$?

Let $X$ be a Banach space. $T(t)$ a family of bounded operators for $t\in\mathbb{R}$. $T(t)^*$ is the adjoint operator of $T(t)$. If $\sup_t |T(t)^*|<+\infty$ , then by Hahn-Banach, there's a ...
2
votes
1answer
33 views

uniform continuity of the function $t\mapsto\langle x^*,f(t)\rangle$

Let $X$ be a Banach space. $f:\mathbb{R}\to X$ a function. If we have $t\mapsto\langle x^*,f(t)\rangle$ uniformly continuous on $\mathbb{R}$ for each $x^*\in D$ where $D$ is a dense subset of $X^*$ ...
2
votes
2answers
69 views

If we remove finite number of elements from a dense subset, will it be still dense?

Let $D$ be a dense susbet of $X$. If we remove a finite number of elements from $D$, will it still be dense in $X$?
1
vote
1answer
60 views

Weak* compactness of the unit ball

Things that we know: In any topological space compactness implies sequential compactness If E is any topological space the then the closed unit ball $$ B_E=\{f\in E^*; \|f\|\leq 1\} $$ is compact ...
5
votes
1answer
104 views

Connection between separable measure spaces and $\sigma$-finite measure spaces

I recently came across a theorem which makes a hypothesis that a certain measure space is separable (the definition can be found here). In order to avoid confusion, I'll add the definition here: We ...
10
votes
1answer
239 views

What is the Atiyah-Singer index theorem about?

I was just a little bit curious about the general statement of this theorem. Honestly, I am not at all interested in fully understanding this, so it is not that I am too lazy to read plenty of books ...
4
votes
1answer
59 views

Weak topology is not metrizable: What's wrong with this proof

Let $(X,\|\cdot\|)$ be a infinite dimensional normed vector space, and Suppose that the weak topology in $X$ is metrizable by a metric $d$. How the opens of $\tau_d $ should be the same as the ...
0
votes
1answer
55 views

Why the weak topology and the strong topology coincide?

Why in any finite-dimensional Hilbert space the weak topology and the strong topology coincide?
4
votes
2answers
67 views

Operators $A$ such that $e^A$ is norm preserving

Let $X$ be a Banach space. $A$ a bounded operator. We can define the exponential of $A$ by $$e^{A}=\sum_{n=0}^{+\infty}\frac{A^n}{n!},$$ which is also a bounded operator. Is there any sufficient ...
5
votes
2answers
44 views

Is the set of translations of a function compact?

Let $X=BUC(\mathbb{R})$ be the Banach space of real bounded uniformly continuous functions on $\mathbb{R}$ equipped with the supremum norm. Let $f\in X$, then the subset $$\{f_a:t\mapsto f(t+a), \ \ ...
1
vote
1answer
64 views

Why $(X,d)$ is a complete $\mathbb{R}$-tree?

Definition. An $\mathbb{R}$-tree is a metric space $(X,d)$ such that there is a unique geodesic segment (denoted $[x,y]$) joining each pair of points $x,y \in X$; if $[x,y] \cap [y,z] = ...
2
votes
1answer
43 views

question concerning weak star convergence.

Given: X seperable Banach space. X' its dual. We have $M\subset X'$ a closed unit ball in X'. Choose a sequence $(x_{n}$) of nonzero elements in X which is dense in X. Define ...
2
votes
2answers
99 views

Is every continuous function measurable?

In non-Hausdorff topology it is standard to define the Borel algebra of a topological space $X$ as the $\sigma$-algebra generated by the open subsets and the compact saturated subsets. Recall that a ...
1
vote
0answers
26 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
3
votes
1answer
38 views

The open sets in Banach space

Let $X$ and $Y$ be two Banach spaces. And we set $X\times Y=\{(x, y): x\in X$ and $ y\in Y\}$. If we take a open set $U$ in $X\times Y$, then does $U$ has the form $U_{X}\times U_{Y}$? Here $U_{X}$ ...
1
vote
2answers
41 views

Constructing a sequence that is pointwise bounded but not uniformly bounded by points in a closed, nowhere dense set in $\mathbb{R}$.

I believe that this question below is asking for a sequence of functions that are bounded pointwise in $\mathbb{R}$ but NOT uniformly bounded in a closed, nowhere dense set of $\mathbb{R}$. Suppose ...
0
votes
0answers
39 views

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous?

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous? I am trying to prove continuity by limits but am failing: suppose ...
1
vote
0answers
41 views

Open ball in infinite dimensional Banach space is not weakly open

I have to prove that open ball in infinite dimensional Banach space is not weakly open. I have no idea how can I do it. I think that I should reach contradiction with infinite dimensions.
0
votes
1answer
69 views
2
votes
1answer
61 views

Countable union of relatively compact sets

Let $X$ be a topological space and $\mathcal K(X)$ be $\sigma$-algebra, generated by compacts of $X$. Prove that for any set $B \in \mathcal K(X)$ either $B$ or its complement can be represented as a ...
1
vote
2answers
70 views

Prove that a closed unit ball in $C[0,1]$ is not weak-compact

I have to prove that a closed unit ball in $C[0,1]$ is not weak-compact. The hint is that I should consider sets: $$V_t=\{f\in C[0,1]:|f(t)|>1/3\}$$ and $$U_t=\{f\in C[0,1]:|f(t)|<2/3\}$$ Now I ...
1
vote
1answer
38 views

Compact surjective non injective operator

Let $X$ be an infinite dimensional Banach space. I know that every compact operator $A$ is not bijective or $0\in\sigma(A)$. Fox example the compact operator $A$ defined on $X=C([0,1],\mathbb{R})$ ...
2
votes
1answer
37 views

A soft question on the dimension of normed spaces

There's some properties such that if satisfied by a normed space, then necessarily this normed space is finite dimensional. An example is of course the compactness of closed bounded sets. Another ...
1
vote
0answers
12 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
1
vote
0answers
21 views

symmetric quasi-uniformity

A quasi-uniformity $U$ will be called symmetric provided that $U = U^{-1}$, that is, provided that it is a uniformity. Otherwise it will be called nonsymmetric. It is readily seen that the supremum ...
0
votes
1answer
41 views

How the weak convergence is related with trivial topology?

I hope my question is not going to be silly, but I am really confused, and will appreciate any help. How the weak convergence is related with trivial topology? Weak convergence and pseudo-metric ...
5
votes
1answer
68 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
0
votes
2answers
42 views

Density of union of closed sets

Let $X$ be a separable, infinite dimensional Banach space. I want to find a countable collection $(V_n)$ of sets in $X$ that satisfy the following properties: For each $n$, $V_n$ is closed and norm ...