29 views

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 $\Lambda = \{n + m\tau :... 1answer 124 views ### Is there a sum of an uncountable set of Real numbers? [duplicate] The addition of Real numbers is commutative, so instead of saying we can find the sum of a sequence$\{a_1,...,a_n\}$of real numbers that are pairwise not equal, we can say that there is a sum of a ... 1answer 581 views ### Is Lebesgue integral w.r.t. counting measure the same thing as sum (on an arbitrary set)? TL:DR; For arbitrary sets (not necessarily countable) we have two notions: Lebesgue integral w.r.t. the counting measure and sum of family indexed by this set. Are these two notions equivalent? For ... 1answer 135 views ### Is there a rigorous way to define uncountable products? I'm dreaming of a way to define an uncountable product of real numbers. Of course any sensible definition should only converge for a sequence with only finitely many terms outside$[0, 1]$. It ... 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 $$S = \sum_{n=0,1,2,...}... 2answers 288 views ### Sum of a series indexed by ordinals If \mu is an ordinal, how can we formalize that$$ \sum_{\lambda<\mu}x_{\lambda}=z $$When \mu=\omega, this is just the usual infinite series, the partial sums converge to z. What is the ... 4answers 827 views ### Why “countability” in definition of Lebesgue measures? According to Wikipedia, the definition of the Lebesgue outer measure of a set E is as follows:$$ \lambda^*(E) = \operatorname{inf} \left\{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{... 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$f$... 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$X$. ... 2answers 121 views ### Positive Sums: Product Lemma for: Trace Positive Sum Given the TAS$\overline{\mathbb{R}}_+$. For product sums: $$\omega:I\times J\to\overline{\mathbb{R}}_+:\quad{\sum}_{I\times J}\omega={\sum}_J{\sum}_I\omega$$ (... 4answers 3k views ### Can we add an uncountable number of positive elements, and can this sum be finite? Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ... 2answers 812 views ### Why do we distinguish between infinite cardinalities but not between infinite values? More specifically, why are we "allowed" to denote$|\mathbb{N}|<|\mathbb{R}|$but not$\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$? Can we distinguish between "countable ... 2answers 1k views ### What is the sum of all real numbers from$0$to$1$? [closed] I wanted to know the approximate sum of real numbers from 0 to 1. Please tell me how we can find it. 1answer 590 views ### Is every Hilbert space an$L^2$space? Let$H$be any Hilbert space. Must there exist a measure space$(X,\scr{M},\mu)$such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you 6answers 2k views ### Is there a number that's right in the middle of this interval$(0, 1)$? This might seem like a silly question, but is there a number that's right in the middle of this interval$(0, 1)$? And the half-open intervals:$(0, 1]$,$[0, 1)$? I know for a fully closed interval$...

15 30 50 per page