Let $ X $ be Banach space, and $X^*$ its dual. A set $ F \subset X ^ * $ is weakly-* compact if and only if $ F $ is closed in the weak* topology and is bounded in norm. How does one prove this (well-known) fact? Notes This characterization of compactness in weak* […]

I just read about one point compactification and i am having some difficulty in grasping the concept. Does one point compactification mean that we are simply adding a point to a non compact space to make it compact. For example, my book says that $S^n$ is the one point compactification of $\mathbb R^n$, i don’t […]

Problem: What is the one point compactification $X^*$ of a discrete space $X$. In the case of $X$ being finite, $X$ itself is compact so the one point compactification would be merely $X$ $\bigcup$ {$\infty$}. Now in the case of $X$ being infinite I need to consider two cases. When $X$ is countably infinite, I […]

The following statement doesn’t make sense to me, can someone justify it to me ? If $K$ is a compact subset of a Banach space $Y$ then there exists for $\epsilon > 0 $ a finite dimensional subspace $Y’$ of $Y$ such that $d(x, Y’) < \epsilon $ for every $x \in K $ , […]

The Kreinâ€“Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is there a “simple argument” that the extreme points is not empty that avoids going through the full Krein-Milman theorem?

I can show using compactness theorem (or even EF Games) that even number of nodes in graphs is not axiomatizable. However, I don’t understand some thing: We can express cardinality of universum, so we should be able to express number of nodes with infinity set of formulas (formula for each even number expressing cardinality of […]

I am looking for a direct proof (not by contradiction) that a compact metric space is sequentially compact, ie constructing a converging subsequence from any sequence. Thanks

I go through a proof of the following. Let $(\ell_1,d)$ be the metric space of all sequences $x = (\xi_i)_{i \in \mathbb{N}}$ with $\sum_{i=1}^{\infty} |\xi_i| < \infty$ and the metric $$ d(x,y) = \sum_{i=1}^{\infty} |\xi_i – \eta_i|, \qquad x = (\xi_i), y = (\eta_i). $$ Theorem: A subset $M$ of $l_1$ is totally bounded (pre-compact) […]

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a compact $G_\delta$ with respect to $F$. Clearly $K$ is compact as a subset of $X$. Must it […]

given: $(X,d)$ is a compact metric space $T:X\to X$ is such that $d(T(x),T(y))<d(x,y)\ \forall x,y\in X$ with $x\neq y$. Prove that T has a unique fixed point. Attempt: I think I can prove Uniqueness: Consider $T(x)=x$ and $T(y)=y,\ x\neq y$ Then, $d(x,y)=d(T(x),T(y))<d(x,y)$ (contradicting original condition). However, I am having trouble executing the proof that the […]

Intereting Posts

Generating functions of partition numbers
Lattice of Gauss and Eisenstein Integers
When does pointwise convergence imply uniform convergence?
Probability of 20 consecutive success in 100 runs.
Prove an inequality
How can it be meaningful to add a discrete random variable to a continuous random variable while they are functions over different sample spaces?
Looking for Proofs Of Basic Properties Of Real Numbers
Approximating $\arctan x$ for large $|x|$
Why does having fewer open sets make more sets compact?
In how many different ways can boys and girls sit a desks such that at each desk only one girl and one boy sits?
Why is a function at sharp point not differentiable?
Questions concerning a proof that $\mathcal{D}$ is dense in $\mathcal{S}$.
Online MathJaX editor
How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$
Weak-to-weak continuous operator which is not norm-continuous