Recent content by QuantumSpace

Why is 14 not solved yet?

Theoretical fluid dynamics does sound like something useful for space industry, but I'm by no means an expert.

My solution for exercise 3. Remark: the same proof works to show that any positive map between ##C^*##-algebras is bounded. That's why I denote the absolute value on ##\mathbb{C}## by ##\|\cdot\|## as well.

Attempt for an elementary solution for (3):

Also, you can here appeal to the fact that countable union of countable sets is countable, if you are ok with ##|\mathbb{Z}_n[X]| = \aleph_0##. Maybe you are right and I overcomplicate the issue.

Are you telling me that I cannot use the standard cardinal arithmetic inequalities ##|\bigcup_{i \in I} A_i| \le \sum_{i \in I}|A_i| \le |I| \sup_{i \in I} |A_i|##? Because really that's all I'm using, together with the fact that ##\aleph_0 \aleph_0 = \aleph_0##. Cardinal arithmetic is invented...

##|\mathbb{Z}[X]| = \aleph_0## is quite easy to prove. Indeed, ##\mathbb{Z}[X]= \bigcup_{n=0}^\infty \mathbb{Z}_n[X]## so since ##\mathbb{Z}_n[X] \cong \mathbb{Z}^{n+1} \cong \mathbb{N}##, we find $$|\mathbb{Z}[X]| \le \sum_{n=0}^\infty |\mathbb{Z}_n[X]| \le \sum_{n=0}^\infty \aleph_0 =...

About question (1) I am a little bit confused. Isn't completeness invariant under changing to an equivalent metric? If so, why would you want to change to an equivalent metric?

(3).

You can use it to give an easy proof of the Banach-Alaoglu theorem, which is absolutely fundamental in abstract functional analysis. And as you said, it can also be used to show that certain compactifications, such as the Stone–Čech compactification, exist.