@Liad @Semiclassical It seems to be saying this: Let $k:[0,1]\to[0,\infty)$ be a non-negative measurable function. Let $X$ be the set of equivalence classes of functions for which that integral [1] exists, modulo a.e. equality. (a) show that $X$ is a vector space. (b) show that the following formula [2] fulfills all the requirements of an inner product, except that $f \ne 0$ might occur even when $(f,f) = 0$.
let $k:[0,1] \to [0\infty)$ be measurable and let $X$ be all the function (module equality a.e) that $\int_0^1 |(fx)|^2 k(x) dx \lt \infty$. i want to find $f \ne 0$ s.t $\int_0^1 |(f(x))|^2 k(x)dx =0$, someone can help?
so @Semiclassical, to be fair, I think we just misinterpreted the quantifiers
@TedShifrin A bi-invariant metric on $G$ of volume 1 is unique up to isometry. What can we say about 'bi-invariant metrics' on orbits? Precisely, this means that it is invariant under the left action of $G/H$ as well as the right action of $W(H) = N_G(H)/H$.
In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.The adjugate has sometimes been called the "adjoint", but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.
== Definition ==
The adjugate of A is the transpose of the cofactor matrix C of A,
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If a question you have posted has "grown" into a series of quite similar questions, but none the less, more than one mathematics problem that requires rigorous proof, is it better that A) I continue to update the original question, by including all the material involved and how it relates to the original to the best of my ability at the present time or B) make a new question for each individual problem, and link them in that feature provided
i think the question is how distinct it is from the initial question
if the scenario were "this is my initial question, but here's the question I should have asked" then I'd lean towards keeping the question
on the other hand, if it's a new question that's motivated by the old one then I'd create a new one
but I'd still try to keep the new question as self-contained as possible. referencing the old question as context is fine, but typically you want the question to make sense by itself
Ok well that in itself is most definiately the problem, but I go thru phases of which particular problem I want the reader to see as central or the most significant, is there anyway I can have a webspace where by it is available to the people I know will provide the right type of criticism necessary,
and then once I have their approval of the content, post the question on SE officially? it's just that atm I am doing choice A, but editing and updating my current position on a public post is a little embarrassing in that its just exposing the lunatic I actually am logic wise, and so it would feel a lot better if I could get help with writing it by having it somewhere number theory people can see, then post a polished "end product"
This sandbox is intended for saving drafts of long, complex posts, especially posts whose composition takes a long time. It serves to localize to one thread the front-page "bumps" caused by edits to drafts of such posts, so that they may be easily ignored. Also, it helps to guard against losing l...
Having a little bit of difficulty interpreting this statement: "Let $\tau$ be a topology on $\mathbb{N}$ with the property that each open cover of $\mathbb{N}$ has a subcover of no more than 2 sets."
@Rithaniel if $\{U_i\}_{i \in I}$ is a family of open sets with $\bigcup U_i = \Bbb N$ then there is $i, j \in I$ with $U_i \cup U_j = \Bbb N$ or there is $i \in I$ with $U_i = \Bbb N$
Okay, so every point "splits" $\mathbb{N}$ into two open sets, and dividing these sets cannot possibly cover $\mathbb{N}$ (therefore, splitting the sets in the subcover must result in at least one non-open set?)
If a question you ask requires both the use of set builder notation and expressions of the fractional part, (traditionally having the notation of curly brackets as used in set builder notation) is it necessary to re-express things in terms of the ${\{x}\}=x-\lfloor x \rfloor$ or something of this nature?
also @Semiclassical do I just post the question as a comment on the page you linked to add it to the "sand box"?
I really dislike the word "plugging" when it comes to mathematics. Is it just me or every time someone says "I tried plugging [some value] into [some function]" it means they don't quite understand what they are doing? Maybe it's just the language barrier for me, I hope I don't offend anyone
user280247
Different numeric systems (decimal, dozens, binary etc) woudn't change anything about calculus, but are there some numeric systems which actually do it?
user280247
For example, numeric system with different lenghts between numbers 0,1,2,3
@santimirandarp Don't quote me on this, but I believe that you can do something like calculus on the p-adic numbers, which have a different notion of distance than the standard Euclidean one.
user280247
hmm interesting, I suppose anyone'd answer. But isn't what I said a possible definition of a function?
user280247
For example, if we plot x axis and below a y axis; which is f(x)
Not sure. As I said, it's likely just the language barrier. Technically, I would use "evaluating the function at the point", but again, I have learned in Russian, so what do I know
user280247
Maybe @YuriyS is referring to a pattern: usually the word appears when less understanding of maths is observed...
If $\{f_n\}$ is a sequence of continuously differentiable functions on $[a,b]$, $f_n \to f$ uniformly on $[a,b[$, and there is a function $g : [a,b] \to \Bbb{R}$ such that $f'_n \to g$ uniformly on $[a,b]$, the $f$ is continuous differentiable and $f'=g$.
So, when I first approached this pr...
Preface:
This question is based on the answer to this question given in the comments.
The problem:
Consider the Lebesgue probability space on the interval $[0,1)$. (I.e. the state space is $Ω = [0, 1)$, the $\sigma$-field is the set of Lebesgue measurable sets and the measure is the Lebesgue...
Does anyone else feel like they go thru an infinite loop of sulking because no one seems interested in the problems they are working on, then realising everyone has their own choices and how bad it would be to force everyone to be interested in one persons problems, then somehow ending up sulking again because no one gives a #$#$ about your specific math problem on this day?