If $A \in M_{n}(\Bbb{R})$ then can we compare the values of $e_{1}^{T} adj(A^2+A+I) A u$ and $e_{1}^{T} (A+I) adj(A^2+A+I) u$, where $e_{1}^{T}$ is a $1 \times n $ standard unit vector, $\xi$ is a $n \times 1$ column vector.
What if $A = \left[C_{1}\middle|\frac{I}{0\dots0}\right]$
I just saw someone claim that if a well-ordered set has no maximum then neither does the set of its limit points. But natural numbers are an easy counter-example
Found it: “ I warn the reader of a personal quirk. I'm told that proper usage requires the addition of a period in a sentence that ends with a set-out equation. But I find extra dots in such equations confusing, so I don't use punctuation in set-out formulae, even if proper grammar says they should be there.“ (from volume one of his text on OPUC)
I had to look it up to make sure that I was remembering correctly, but the official AMS style guide indicates that displayed equations should be appropriately punctuated, and that copy editors should add punctuation if it is missing.
Fun fact: Michel taught a course in functional analysis out of Reed and Simon's book. An academic sibling of mine picked up a cheap used copy. The text was marked to hell, which this person found quite annoying until they tracked down the "ex libris" from the front page. The annotations were written by Wightman (Simon's PhD advisor).
Sure, but when you are making a lot of estimates, the difference between "nonnegative" and "positive" is often not terribly important, and phrases like "monotone nondecreasing" are painful when compared to "monotone increasing"---to maintain the consistency of the language "increasing : positive :: strictly increasing : strictly positive."
(1) I don't think that this convention is "insane" and (2) I genuinely don't think that it is likely to lead to much confusion, particularly considering that it occurs in five volume treatise in analysis which is intended more as a reference than as a teaching tool.
It is a work on analysis (where exact equality is rare, hence the distinction between $(0,\infty)$ and $[0,\infty)$ is largely irrelevant), and it is something that appears in the exposition, not in the actual statements of results (where notation is used).
This might be obvious but for ambient space $M$ and hypersurface $\bar{M}$ of $M$, when defining Second Fundamental Form H(X,Y) = (\nabla_X Y, -n) where \nabla is on M, what is the inner product from? Every book writes it as just inner product, is it metric from $M$ or $\bar{M}$?
No clue. Currently he's an emeritus at Illinois it seems
The figure-eight picture above is originally by Thurston :) Everyone, including expert knot theorists like Rolfsen, was shocked when he first showed that the figure-eight complement is hyperbolizable
If $d$ is a metric then am I right in saying I can use cauchy schwartz to write $d^2(x_1,x_k)\leq \big( \sum_{n=1}^{n=k}d(x_n,x_{n+1}) \big)^2\leq k \sum_{n=1}^{n=k} d^2(x_n,x_{n+1})$ ?
since if we define $d(x_n,x_{n+1})=:a_n$ and $a:=(a_1,\ldots,a_n)$ so $ \big(\sum_{n=1}^{k} d(x_n,x_{n+1})\big)^2=(\langle 1, a \rangle )^2 \leq \|1\|^2 \|a\|^2$
But there is a distinction, those two things are entirely different apparently. Ross Millikan's answer (highestest voted) suggests that it is about sequences
I am reading the limit point of sequence and limit of sequence. I can't understand the difference between limit point of sequence and limit of sequence. In my book they told me following two points.
"If l is the limit point of $\{x_n\}$ , then every nbd of l containing an infinite number of its...
And the question you just cited is not relevant, as it is about the difference between the limit (of a sequence) and a limit point of a set (where that set happens to be the set of terms in a sequence).
Basically an accumulation point has lots of the points in the series near it. A limit point has all (after some finite number) of the points near it.
Think of the series $(-1+\frac 1{n^3})^n$. Both $-1$ and $1$ are accumulation points as there are entries very far out close to each. Neither is...
The issue is like when I found the post I was entirely confused because the answer and post was at a disconnect (yet answer is most upvoted). It should be either that question is modified or the answer is changed
I kept looping for about 10-20 mins being confused on what is the correct definition, so I thought this may save time for some one else
@Beautifullyirrational Unfortunately, such answers (that are only correct AFTER the edit) are often upvoted to heaven , and then it is difficult to get rid of them.
damn this was a very damagin answer by Ross tho because this question was viewed 41 k times xD
Okay, I'll go now but please edit the question title to be that of limit point & accumulation point in context of sets because it is very confusing as is.
As far as I understand there are two type of limit point, one of set and one of sequence and the limit point of set. In sequence case limit point = limit itself. Here, the OP is asking about sets as we can say they have mentioned sets in the body. So, I don't understand why you say my reading is off
what is user87690 "Note that these are notions of accumulation point and limit point of a sequence rather than of a set, which is related but different things" saying here?
My objection to Ross Milikan's answer is that it only considered sets which occur as sequences. The answer refers only to the special case where one is interested in the limit points of a sequence. I understand the comment to be pointing out that this is a special case of something more general.
Though, to be clear, the language here is a bit imprecise. A sequence is a function $a : \mathbb{N} \to X$, where $X$ is some set (or, in this case, topological space).
What we are actually talking about is the image of $\mathbb{N}$ with respect to $a$, i.e. the set $a(\mathbb{N})$.
Hence the phrase "accumulation point of a sequence" is a little bit imprecise. A more precise statement would be "accumulation point of the image of a sequence". But this is the kind of precision which is often elided, as it rarely causes confusion (as we tend to forget fairly early on that sequences are functions).
in the definition of accumulation point, do we just need that open nbhd about a point's intersection with ambient set is non empty or does it need infinity of points?
") an accumulation point of a set is a point, every neighborhood of which contains infinitely many points of the set."
@Beautifullyirrational A point $x$ is an accumulation point of $X$ if every open set containing $x$ contains a point of $A$ other than $x$. There are examples of spaces where the topology is such that any particular open set might only contain one other point.
As a dumb example, consider the real line, plus an extra point $\ast$. Take the open sets here to be the open sets of $\mathbb{R}$, and the set $\{\ast,0\}$ (and sets of the form $U\cup \{\ast,0\}$ with $U$ open in $\mathbb{R}$, since we need this to make sure that we have a topology).
That post is incorrect, but is basically correct in the context of sets of real numbers. The question is tagged real-analysis, so it is not too wrong for the given context.
Oh, that answer also defines a limit point incorrectly.
@Beautifullyirrational In $\mathbb{R}$, a neighborhood of a limit point of a set $A$ will contain infinitely many members of $A$. But this follows from the structure of $\mathbb{R}$.
@Beautifullyirrational Off the top of my head? I have no idea. I tend not to care too much about these kinds of nitty-gritty topological details. Also, the counter-examples that I can think of off the top of my head are generally not metrizable.
There is also the issue that the terms "limit point", "accumulation point", and "cluster point" are used interchangeably by some authors, and mean distinct things to other authors.
It is important to know what definitions the text you are reading is using.
The difference is very simple.
1) As you wrote: an accumulation point of a set is a point, every neighborhood of which contains infinitely many points of the set.
2) But a limit point is a special accumulation point. No matter how small neighborhood you choose, all members $a_n$ (after a certa...
@XanderHenderson how would you define a limit point?
As I said above: "There is also the issue that the terms "limit point", "accumulation point", and "cluster point" are used interchangeably by some authors, and mean distinct things to other authors."
If you are referring to "But a limit point is a special accumulation point. No matter how small neighborhood you choose, all members 𝑎𝑛 (after a certain 𝑛) are in the neighborhood of the limit point," I would say that this is an informal description of the limit of sequence.
For the record, Munkres does not distinguish between the terms "limit point", "cluster point", and "point of accumulation". He defines a limit point of a set $A \subseteq X$ to be a point $x$ such that $(A\cap U) \setminus \{x\} \ne \varnothing$ for all open sets $U\subseteq X$ which contain $x$.
Oh, I was trying to understand compactness equivalent to Bolzano Weierstrass proof, in that one of the point mentioned was if sequence has convergent subsequence that it has limit point or something of that sort
I mean the idea I had was the sequences are a bit more discrete, so unless the sequence is constant, it should usually be that not every nbhd centered at point of sequence has point other than itself
but if you have a like a connected set like R^2 (with some norm metric), then it feels like each point at R has a lot of points other than itself no matter what nbhd we take
@Beautifullyirrational The rational numbers are countable, which means that there is a function $q : \mathbb{N} \to \mathbb{Q}$ which is bijective. This function is a sequence. Every element of $\mathbb{R}$ is a limit (cluster, accumulation) point of this sequence.
how it can be that every element is? suppose we have a point far away from where the stuff with the sequence is going (imagine sequence is hopping around in some bounded set), if we take a faraway point outside this circle which bounds our set, it should not be that any open set containing this point has points from the sequence you have given
@Beautifullyirrational For many reasons, yes. The next issue you need to tackle is "What do you mean by 'more'?" And, again, you need to be precise about your definitions.
For example, the Wikipedia article has a discussion on the accumulation points of a sequence which offers definitions that I have never seen before, nor had cause to care about.
Oh yeah, haha, there were a lot of new words there for me as well
This type of issue I saw in ring theory/ field theory as well. I wish there was like an international council of math to fix conventions in a certain way (ppl do this already for languages)
To standardize all of mathematics, you would first need to be fairly intimately familiar with all of mathematics. And if you attempted to introduce a new idea, you would first need to read all of mathematics to ensure that your notation and definitions are totally distinct from what anyone else has ever done.
Otherwise, you run the risk of overloading terms.
Also, the French already attempt to do this for French. They largely fail. That isn't how languages work.
High German is a German dialect which is the version of German generally taught to foreigners, and is typically understood by anyone who speaks any form of German.
But, to my knowledge, there is not a "German Academy" which seeks to standardize and fix this version of German for all time.
It is still a natural human language, subject to the evolution of natural human languages.
@Beautifullyirrational What is your definition of "barbecue"? My understanding is that barbecue involves a sauce. I'm just grilling ground lamb and peppers.
I am reading the limit point of sequence and limit of sequence. I can't understand the difference between limit point of sequence and limit of sequence. In my book they told me following two points.
"If l is the limit point of $\{x_n\}$ , then every nbd of l containing an infinite number of its...
Basically an accumulation point has lots of the points in the series near it. A limit point has all (after some finite number) of the points near it.
Think of the series $(-1+\frac 1{n^3})^n$. Both $-1$ and $1$ are accumulation points as there are entries very far out close to each. Neither is...
The difference is very simple.
1) As you wrote: an accumulation point of a set is a point, every neighborhood of which contains infinitely many points of the set.
2) But a limit point is a special accumulation point. No matter how small neighborhood you choose, all members $a_n$ (after a certa...
