Linked Questions

2
votes
3answers
404 views

Can absolute convergent series be expressed as sum of two series?

Let $C\subset \omega \bigwedge A\bigcap B = \emptyset \bigwedge A\bigcup B = C$. Let $\{x_i\}$ be a sequence of nonnegative reals. Suppose $C$ is infinite and $\sum_{i\in C} x_i$ converges. (Since ...
5
votes
1answer
300 views

Convergence of Series of a Net's terms

I'm working through Dr. Pete Clark's convergence notes here: http://math.uga.edu/~pete/convergence.pdf and I've been thinking about Exercise 3.2.2 (a) and I am completely stumped. The exercise says ...
7
votes
0answers
294 views

Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
8
votes
0answers
264 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
2
votes
1answer
142 views

Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.

Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that $\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to $L_2\left(\bigsqcup\{(X_\nu,\mu_\...
0
votes
2answers
121 views

Positive Sums: Product

Lemma for: Trace Positive Sum Given the TAS $\overline{\mathbb{R}}_+$. For product sums: $$\omega:I\times J\to\overline{\mathbb{R}}_+:\quad{\sum}_{I\times J}\omega={\sum}_J{\sum}_I\omega$$ (...
1
vote
1answer
135 views

Is there a rigorous way to define uncountable products?

I'm dreaming of a way to define an uncountable product of real numbers. Of course any sensible definition should only converge for a sequence with only finitely many terms outside $[0, 1]$. It ...
2
votes
1answer
29 views

Meaning of absolute convergence when summing over countably infinite set.

I'm trying to understand the chapter on Elliptic functions in Stein's Complex Analysis. In particular, I am interested in the construction of Weierstrass's $\wp$ function. Let $\Lambda = \{n + m\tau :...
0
votes
0answers
45 views

Uncountably infinite series [duplicate]

I was fascinated when I heard that the most intuitive laws of arithmetic are no longer necessarily valid when it comes to the sum of an infinite sequence, which be denoted by $$S = \sum_{n=0,1,2,...}...

15 30 50 per page