## Rephrasing the definition of a limit

Is lim The same as saying “one can get \large{f(x)} as close as imaginable to \large{L}, by setting \large{x} close enough to \large{a}” and vice versa “one can get \large{x} as imaginable to \large{a} by setting \large{f(x)} close enough to \large{L}“? Sorry if obvious questions, but I need to understand this before i can grasp … Read more

## Can anyone please help to clarify the sentences ” into a fat tail part in L2 plus a fat body part in L1.”

In the link https://en.wikipedia.org/wiki/Fourier_transform#On_Lp_spaces what does this sentences mean: into a fat tail part in L2 plus a fat body part in L1? Would anyone please help? Answer The idea is that you first define the Fourier transform on L1 and L2. Then for f∈Lp for some p∈(1,2), write f(x)=1{|f(x)|≥1}f(x)⏟f1(x)+1{|f(x)|<1}f(x)⏟f2(x). Then f1∈L1 is the tail … Read more

## Continuity Help

If lim for every x \in \Bbb R, does it follow that f is continuous? I start by rearranging it to be \lim_{h \to 0}f(x+h)+f(x-h)=\lim_{h \to 0}2f(x) and I feel as though I need to rearrange the LHS, but am unsure as to how to proceed. Thank you in advance Answer Counterexample: f(x) = \begin{cases} … Read more

## Is a non-principal ultrafilter the same thing as a free ultrafilter?

Can someone please confirm if a non-principal ultrafilter is the same thing as a free ultrafilter. I keep finding conflicting definitions so am not sure. Answer They’re the same. Sometimes “free” is simply defined to mean “nonprincipal”. Another definition given is: An ultrafilter $\mathcal{F}$ is free if $\bigcap \mathcal{F} = \emptyset$. But that definition is … Read more

## What does bounded in module mean?

I’m trying to understand Liouville’s theorem and in the book I’m reading it’s stated as Let f be a holomorphic function on the complex plane, which is bounded in module in a neighborhood of infinity (i.e. bounded in module in all of C), then f is a constant function. What exactly does “bounded in module” … Read more

## Problems with the definition of vectors as directed line segments in $\mathbb{R}^3$

First I’ll say where I’m working: The vectorial spaces $\mathbb{R}^2$ and $\mathbb{R}^3$. Then I’ll define a vector of this spaces as the following: $\textbf{Definition. }$ A vector $\vec{v}$ is the set of all equal directed line segments. Now suppose that $$\underbrace{\overrightarrow{AB}}_{\mbox{directed line segment}} \in \vec{v},$$ which is a correct notation, by definition. So why do … Read more

## Name of integers whose prime factorization is exponent free [squarefree]

Let n>1 be a positive integer. Suppose that n is a product of distinct primes n=k∏i=1pαii i.e. αi=1 for every i. Said equivalently: n has no repeated (square) factor. Surely if n=pq we call them semiprimes but beyond that do we have a general term? Would you just say that n is a “product of … Read more

## What does an asterisk mean in analysis?

Please help me to identify what the asterisk symbolizes in the following: I would have thought it would mean a dual space, but the author uses ′ to symbolize the dual space. In fact, the preceding paragraph is this: If it is the dual space, I do not understand how lq is the dual of … Read more

## Connection Between Moreau Envelope and the Conjugate

Let f be a proper, lower semicontinuous function, then the Moreau envelope is given by: eλg(x):=infy{g(y)+12λ‖ Recall that the Fenchel conjugate is: Given f: \mathbb{R}^n \to \mathbb{R} f^*(x) := \sup_y \{x^Ty – f(y)\} Are these two definitions connected somehow? It seems that if g(y) = -x^Ty, and f(y) = \dfrac{1}{2\lambda}\|x-y\|^2, the two definitions will coincide. … Read more

## A weakening of the topology axioms

A topology is a set of subsets satisfying certain axioms, including closure under arbitrary unions and finite intersections. Is there a term for the weakened structure given by changing closure under arbitrary unions to finite unions? Answer The structure is just a lattice of sets, albeit with the restriction that it must contain the empty … Read more