# Talk:Mathematical analysis

## Subdivisions

Among the subdivisions, shouldn't classical analysis of metric spaces be listed? Daphne A 15:43, 5 February 2006 (UTC)

Don't know. Is there indeed very serious analysis done on metric spaces? All I know is that people do the minimum necessary for studying topology. Anybody willing to write an article about analysis on metric spaces? Then we will see better. :) Oleg Alexandrov (talk) 16:37, 5 February 2006 (UTC)
Aren't there some simple fixed point theorems (whose names I have forgotten) that apply to metric spaces in general, which are used for things like proving existence of solutions of DE's? Dmharvey 17:07, 5 February 2006 (UTC)
Analysis of metric spaces contains important theorems that have wide applicability (a common text is Marsden's Elementary Classical Analysis). For the nonce, I've amended the opening sentence to indicate that the list of subdivisions is not complete. Hope that's okay—  Daphne A 11:31, 6 February 2006 (UTC)

Subfield Differential Geometry says "the application of calculus to abstract mathematical spaces that possess a complicated internal structure". I would write "that possess a specific internal structure" instead, as there won't ever be an agreement, what is complicated and what isn't. —Preceding unsigned comment added by 81.62.141.37 (talk) 20:41, 8 May 2009 (UTC)

## Mathematical analysis removed from Wikipedia:Good articles

Template:La was formerly listed as a good article, but was removed from the listing because it lacks references, unfortunately. —Preceding unsigned comment added by Worldtraveller (talkcontribs) 18:17, May 22, 2006

## A branch or any branch?

The intro states: "Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence." Isn't it simpler to say that it is a branch that is based on these concepts, and that it has several subfields? --LambiamTalk 08:30, 18 June 2006 (UTC)

I'm not sure I like the description in the article, but referring to it as "a branch of mathematics" is inaccurate, too.
While this doesn't concern people so much these days, attempts have been made to separate the entirety of mathematics into analysis, algebra, and geometry (or a similar small number of superbranches). Today, while many mathematicians use little or no analysis, I'd wager a guess that most mathematicians are in fact familiar with it, and occasionally make use of it.
Put slightly differently: it's not like a third of maths uses analysis, a third uses geometry, and a third uses algebra. It's more like three quarters use analysis and three quarters use algebra, with most of current mathematics falling in the overlap.
RandomP 18:31, 18 June 2006 (UTC)

But is it in any way more inaccurate? You too use "analysis" as if it is a defined entity, rather than a generic designation like "religion" is. One doesn't say that someone "belongs to religion"; one says that they "belong to a religion". But one doesn't say that mathematicians "use an analysis", meaning it's harmonic analysis, or functional analysis, or X analysis for some other X. Of course most work in any present-day branch of maths uses concepts and results from other branches, but that does not imply such branches can't be usefully identified as such. --LambiamTalk 12:37, 25 June 2006 (UTC)

True. I'd suggest going with "a branch" for now, until someone comes up with a better way to express this. RandomP 13:24, 25 June 2006 (UTC)

## Indian primacy claim

What is going on here? Did India invent the calculus? Wow. That is great. But I think it is also inaccurate. While India is to be thanked for the invention of zero, no one can take seriously the assertions made in this article about the invention of the calculus. I implore our Indian friends not to falsify history and be content in knowning that while they have done many great things, the calculus is not one of them.—Preceding unsigned comment added by 70.80.248.67 (talk) 08:48, June 25, 2006 (UTC)

See Yuktibhasa for references on this. -- thunderboltza.k.a.Deepu Joseph |TALK04:44, 12 August 2006 (UTC)
Are there any modern references? Texts on the history of mathematics? Thenub314 12:58, 27 September 2006 (UTC)
There is an Easter egg in the sentence ending: "... with the possibly independent invention of calculus by Newton and Leibniz." The phrase "possibly independent" links to Kerala School#Possible transmission of Keralese mathematics to Europe. --Jtir 20:10, 29 September 2006 (UTC)
I took a look at that section but most of the references I trust claim much less then is claimed here. Thenub314 00:26, 30 September 2006 (UTC)
Further, the word possibly is a weasel word. --Jtir 13:01, 30 September 2006 (UTC)
This edit introduced possibly and this one hid the Easter egg. --Jtir 13:30, 30 September 2006 (UTC)

The first reference cited in Kerala School begins: "It is without doubt that ...". With a lead like that, why read further? --Jtir 15:37, 30 September 2006 (UTC)

Maybe, but even with such a lead, the article does not to claim the things here. Yes, that article has a clear perspective on the issue. And the language they use is "the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus", which is not to say they had known calculus. Thenub314 16:20, 30 September 2006 (UTC)

Good find. That is nicely nuanced, although "anticipated" is a bit vague. Thanks for adding it here. --Jtir 16:42, 30 September 2006 (UTC)
I was not intending to trash talk anyone. It was just news to me, I have read a few accounts of the history of mathematics that had never mentioned them. And when I stared reading biographies, they honestly claimed less then this article does. I would just like to see a reference included so interested people, (like myself) can read more about it. Thenub314 (talk) 16:42, 26 April 2008 (UTC)

Indian claim to having invented calculus is baseless. Fact of the matter is Indian society has always been most mystical of all the cultures in the world. I do not say that as a negative thing. Not having clear definitions of good and evil has resulted in a culture that is extremely tolerant. But, it has also developed a language that is pretty much devoid of any set grammatical rules. Even to this day, in Indian languages it is hard to tell the difference between past and present, between genders, singular and plurals, etc.; inflection of voice, tones and context are your only guide to understanding the sentences. Also, Indian religion is the least clearly defined of all religion - that turns out to be a good thing in many ways. However, claiming great developments in mathematics in a culture so averse to hard and fast rules, to sharp boundaries, is inconceivable. Indians are great students, they have learned well from other cultures, but I do not think they have invented much in science or mathematics. I think even the claim to having invented any sort of numbers is a stretch. Giving Indians the credit for having invented zero, is in fact a European, particularly British, act of soft terrorism against the people of middle east, the real inventor of numbers and mathematics. The harshness of Middle Eastern religions, the invention of law and order in Middle East (Hammurabi), all indicate natives' affinity for rules and sharp definitions. Nowadays the subject of history has become a weapon of mass demoralization or a mass motivational therapy. India has taken full advantage of it. Having the middle east stuck in morass of Western disfavor, our Indian friends are busy taking credit for all things invented more than 200 years ago. Take for example the Indian culinary claims. No Indian restaurant would give credit to the Muslim rulers of India for having created all that is known as "Indian food" in the West. Same is true for art, architecture and music. Invention of mathematics is one of these false claims. I think, never mind the differential calculus or numerals, even the zero could not have been invented in India. —Preceding unsigned comment added by 141.202.248.68 (talk) 18:30, 8 September 2009 (UTC)

Delete the section - I have asked in Wikipedia_talk:WikiProject Mathematics#Kerala fundamental contributions to calculus for suggestions on this and also deleting the related articles claiming Indian primacy in calculus. Thanks, Anand (talk page) 03:31, 7 May 2013 (UTC)

## mentioning "infinitesimal analysis" in the history section

There is reason to mention the term "infinitesimal analysis" in the history section.

• "Of the new or infinitesimal analysis, we are to consider Sir Isaac Newton as the first inventor, Leibnitz, a German philosopher, as the second; ..."
• "The fluxionary and differential calculus may be considered two modifications [in the matter of notation] of one general method, aptly distinguished by the name of the infinitesimal analysis."
Professor Playfair's "Dissertation on the Progress of Mathematical and Physical Science"
as quoted by John Spare, The Differential Calculus, Bradley, Dayton and Co., 1865. [1]
• Infinitesimal analysis is "an archaic term for calculus." [2]
• "The name "mathematical analysis" is a short version of the old name of this part of mathematics, "infinitesimal analysis" ; the latter more fully describes the content, but even it is an abbreviation (the name "analysis by means of infinitesimals" would characterize the subject more precisely)."[3]

--Jtir 16:18, 29 September 2006 (UTC)

## definition of numerical analysis

"Numerical analysis, the study of algorithms for approximating the problems of continuous mathematics."

"Numerical analysis solves problems of continuous mathematics by iteratively computing approximations until they converge to a numerical solution."

• It is now a declarative sentence.
• I didn't understand the phrase "approximating the problems", so I replaced it with the "solving problems" formulation.
• It now identifies both the computational and numerical aspects of the subject.
• The word "algorithm" got dropped. Maybe it could be worked in.
• As I read somewhere on WP, some numerical analysts do not actually write or run computational programs themselves. However, running programs is the ultimate objective of the subject. My definition doesn't quite capture these nuances. Maybe saying "analyzes and solves problems" would be broad enough.
• The definition is longer.

--Jtir 17:20, 29 September 2006 (UTC)

The new definition is wrong. Numerical analysis is not merely about iterative methods. Fredrik Johansson 17:39, 29 September 2006 (UTC)
Thanks for pointing this out. The definition does not account for direct methods. I reverted. --Jtir 19:01, 29 September 2006 (UTC)
Please stick to the formulation at Numerical analysis: Numerical analysis [is] the study of approximate methods for the problems of continuous mathematics. Concerning the use of the word "algorithm": Take for example the Runge–Kutta methods. It is strange and in any case unconventional to call these methods "algorithms". (And, while R-K methods are iterative, it is downright wrong to state that they iteratively compute approximations until they converge.) Likewise, interpolation and extrapolation are methods, not by themselves algorithms.
—The preceding unsigned comment was added by Lambiam (talkcontribs) 18:57, 29 September 2006 (UTC)

(I have copied this discussion to Talk:Numerical analysis) --Jtir 19:34, 29 September 2006 (UTC)

## Dedekind

I added some words to the sentence that references the Dedekind Cut. My words were "… a mathematician creates irrational numbers that serve as continuities between discontinuous rational numbers." I based my words on the actual quote by Richard Dedekind, in which he clearly explains how his Cut reflects the continuum of real numbers. These are his own words:

Whenever, then, we have to do with a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ....

— Richard Dedekind, Continuity and Irrational Numbers, Section IV

My words were quickly altered by User:Gandalf61 to "… allow irrational numbers to be developed rigorously from the foundation of rational numbers." In his edit summary, he asserted that "rational numbers are continuous, but not analytically complete." This must be an error, because rational numbers are most decidedly discrete, not continuous. Their discontinuity is the very reason for Dedekind's Cut. It must seem to User:Gandalf61 more acceptable for a mathematician to "develop rigorously" an irrational number to fill in the gaps between the discrete rational numbers. However, Wooster Woodruff Beman, Professor of Mathematics at the University of Michigan, had translated Dedekind's words into English. He chose to have Dedekind's mathematicians "create" new irrational numbers, rather than develop them from foundations. Dedekind's German had "… so erschaffen wir eine neue, eine irrationale Zahl" (emphasis is Dedekind's) with "erschaffen" meaning "create", not "develop". (See http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html#p4) It is my opinion that my words, now deleted, were closer to Dedekind's own words than are those used by User:Gandalf61.Lestrade 21:21, 11 January 2007 (UTC)Lestrade

Hi, Lestrade! How's life at Scotland Yard? :)
I don't want to get in the middle of an argument. But I do want to clear one thing up. Saying that the rational numbers are "not continuous" is imprecise. I think the technical language used to describe the situation is "the elements of ℚ are dense, but the field ℚ is not complete". That is to say, there are no "holes" in ℚ, because no matter how close together two rational numbers are, we can always find infinitely many more rational numbers in between those two. So in that sense the set ℚ is continuous, because no rational number has a "nearest neighbor". But ℚ is not algebraically complete, because we can write simple polynomial equations with rational coefficients whose roots do not lie in the field ℚ: x2 − 2 = 0 is probably the most famous example of such a polynomial equation. It is also possible to construct Cauchy sequences of rational numbers whose limit is not in ℚ – such sequences are equivalent to Dedekind cuts for irrational numbers.
Your point about what Dedekind thought of his own invention is well taken. On the other hand, the Dedekind cut is not the only way to construct the set of real numbers ℝ. Like Wagner, Nietzche, and Schopenhauer, Dedekind had a swelled head. DavidCBryant 22:41, 11 January 2007 (UTC)
Have gone back and checked my copy of Essays on the Theory of Numbers. I think the misunderstanding might be cleared up by the following quote:
"...the creation of new numbers such that the domain of numbers shall gain the same competeness, or as we may say at once, the same continuity, as the straight line".
So Dedekind was using completeness and continuity interchangably, whereas the modern definitions of these terms are distinct. And clearly Dedekind did not literally "create" irrational numbers, because they had been used for hundreds of years before his time. Anyway, can I suggest a compromise - if you wanted to put in a quote from Dedekind, so that it is clear the terminology used is his (or his translator's), and may not correspond to modern terminology, then I would have no problem with that. Gandalf61 00:22, 12 January 2007 (UTC)
Lestrade - thought about this a bit more. You are right - my edit summary was incorrect - the rational numbers are not continuous. I have reverted my change, so your edit has been restored. I still think that if you want to use Dedekind's own words (or his translator's) you should make it clear that this is a quote from Dedekind, and cite the source. But I will let you or someone else sort this out. Gandalf61 10:26, 12 January 2007 (UTC)
The ambiguous word "continuity" can mean: the property of being continuous (for various meanings of continuous), which in the context of mathematics may apply to continuous functions; or: the property of being similar to a continuum, in mathematics more specifically being homeomorphic to a topological continuum. Dedekind applies it to the "domain of numbers", so the latter meaning is applicable (and appropriate for the ensemble of numbers definable by means of a Dedecut). Neither meaning seems applicable to the ensemble of irrational numbers without their rational progenitors, nor to irrational numbers individually. A little problem with the present text, "irrational numbers that serve as continuities", is that it suggests one – in particular the latter – of these not entirely appropriate interpretations.  --LambiamTalk 20:56, 12 January 2007 (UTC)

User:Gandalf61’s assertion that rational numbers are not continuous is brought into question by User:Lambiam’s remark on the ambiguity of the word continuity. Actually, it is impossible to discuss the topic intelligently at all due to the problem of ambiguity. When Murray Gell-Mann chose the word quark, from James Joyce’s novel, to designate his concept of elementary particles, he was being very uncharacteristic. Most scientists and mathematicians do not choose uncommon or new words as signs for their concepts. They chose, instead, words that are in common use and that already stand for other concepts. This is easily demonstrated by their use of words such as ring, field, space, domain, rational, color, flavor, function, gauge, string, set, and many others. This laziness and inconsideration naturally results in ambiguity and unclear thinking. So, the discrete, unique, separate rational numbers on the number line cannot be said to be discontinuous, because the word continuity has many different meanings and designates multiple concepts. Discussion of continuity must take context into account. We are then in a Twilight Zone of Hegelian Logic and double entendres (as well as triple, quadruple, and quintuple entendres) where concepts gradually turn into their opposites and all discussion is in the secret coded language of high priests and their initiates who alone are versed in the esoteric mysteries. 69.19.14.40 21:00, 13 January 2007 (UTC)AubreyAubervilliers

Well, if you say so, Aubrey. I don't think you're right, though. Language is inherently ambiguous, because it is used to symbolize thought. No matter how carefully I choose my words, there's a chance that you will misinterpret what I said, because the concept you associate with a particular symbol I employed is not necessarily the same concept I had in mind when I chose my words. Yeah, there are plenty of weasel words floating around, but "continuity" is not one of them. Saying that the rationals are "separate" makes some sense, if you're thinking about Cantor's transfinite cardinal numbers, or Dedekind's cuts, or simple continued fractions in canonical form. But it doesn't make any sense at all in terms of the natural ordering of the rational numbers, because no rational number has a "nearest" neighbor.
None of this proves anything. It is an example of human limitations, though. We're still struggling with the infinite when we talk about things like continuity. So it's hardly surprising that some confusion persists, in spite of man's best attempts to be precise. And I think it's fundamentally unfair to call a mathematician "lazy" for using a word that already exists when he tries to explain his ideas to others. He's not communicating with the vacuum. He has to think about his audience, and the best way to get his ideas across to other people. DavidCBryant 22:46, 13 January 2007 (UTC)
“No rational number has a ‘nearest’ neighbor.” In speaking of rational numbers, I would think that, on the number line, a rational number’s nearest neighbors are the rational numbers that are positioned to its left and right on the line. For example, the nearest neighbors of the number 3 are the numbers 2 and 4. This may be insultingly clear and may require being translated into recondite technical terms in order to be of interest.66.82.9.49 23:57, 13 January 2007 (UTC)AubreyAubervilliers
Uh, 3 and 4 are integers. Rational numbers are numbers like ½, and ¼, and ¾, and ⅛, and ⅞. I guess a technical definition is that q is a rational number if (and only if)
${\displaystyle q={\frac {m}{n}}\,}$
where m and n are integers, and n ≠ 0 (that is, m and n are either positive or negative counting numbers, or zero ... except that n cannot be zero, because division by zero is impossible). So 3 and 4 are indeed rational numbers. But 3.5 is also a rational number, as is 3.14159265357989 – if it isn't clear yet what I meant when I said “No rational number has a ‘nearest’ neighbor” then I don't suppose there's any point in trying to explain it again. DavidCBryant 02:43, 14 January 2007 (UTC)
Nice and clear explanation. No ambiguity at all. I think that what you meant is clear. It's possible that the concept you have in mind has been communicated so well that other people can also share the same concept. A further explanation, without using the metaphor "neighbor" might be helpful. "Nearest" is an adjective that is related to space, so it is consistent with the use of the number line, a spatial model. By retaining the same number of decimal places, I would say that the nearest neighbors of 3 are 2 and 4. The nearest neighbors of 3.5 are 3.4 and 3.6. The nearest neighbors of 3.14159265357989 are 3.14159265357988 and 3.14159265357990.69.19.14.43 04:28, 14 January 2007 (UTC)AubreyAubervilliers
"The concept you associate with a particular symbol I employed is not necessarily the same concept I had in mind."
Isn’t this a very good reason for scientists and mathematicians to always use, by convention, one word as a sign for one concept?
"He has to think about his audience, and the best way to get his ideas across to other people."
Isn’t this a very good reason for scientists and mathematicians to avoid using one word as a sign for more than one concept?
If so, why is continuity allowed to have multiple meanings? Laziness and lack of consideration or a best attempt to be precise?66.82.9.49 00:18, 14 January 2007 (UTC)AubreyAubervilliers
If we try to develop a concept of nearest neighbour based on decimal expansions, we run into problems with rational numbers which do not have finite decimal expansions. What, for example, is the nearest neighbour of 1/3 ? Same problem occurs if we use a different base, such as binary - some rationals will not have a finite expansion in that base either. Gandalf61 08:47, 14 January 2007 (UTC)

I suspect we are witnessing the manifestation here of the wielding of the obfuscatory powers of permanently banned User:Jon Awbrey. Please ignore the utterings of these sockpuppets. All I wanted to say is that the wording could be improved a little, which I now (hopefully) have done.[4]  --LambiamTalk 10:34, 14 January 2007 (UTC)

There is no "nearest neighbour" to any number! Pick a number, any number, say x. Let's assume that x has a nearest neighbour, say y. Well, let's just choose another number, say z, which is half way between x and y, i.e. z := (x + y)/2. Well clearly z is closer/nearer to x than y is, (mathematically |x - z| < |x - y|). This is a contradiction, and so our assuption that x had a nearest neighbour must have been false, i.e. x does not have a "nearest neighbour". Any arguement against this is little more than semantics.  Δεκλαν Δαφισ   (talk)  20:41, 13 February 2009 (UTC)

## References, and order of presentation

OK, hopefully the silly season is over. Thanks, Lambiam.

I've started to add some references to this article, so we can remove the "unsourced" tag. I'd like to get some feedback on the way I've set it up (with <ref> tags in the body of the article, a "Notes" section for the generated footnotes, and a "References" section that some or all of the "Notes" can point to). Is everybody OK with that? Or is it too verbose?

On a closely related topic, I'm wondering about the order of presentation within the subsection "Subdivisions". Should this be placed in historical order? Or is it OK as it stands? (The main thing that looks out of place to me is "functional analysis" ... the order I'd prefer is real/complex/harmonic/functional/topology/p-adic/nonstandard, which is more or less the historical order of development, AFAIK. Numerical analysis is pretty old, but since it's applied math, putting it last seems appropriate.) What about the style? It seems a bit jumbled to me. Some of the bullet points are complete sentences, and others are not. Should that be straightened out? DavidCBryant 11:47, 14 January 2007 (UTC)

Discussions that are informative and educational are not silly. Not everyone has had the benefit of a CalTech education.66.82.9.58 14:20, 14 January 2007 (UTC)AubreyAubervilliers

I've added two new refererences, Hewitt and Stromberg "Real and Abstract Analysis", and Carl L. Devito, "Functional Analysis". I've also attempted to add some motivation to the introductory section. Greg Woodhouse 22:25, 20 March 2007 (UTC)

## St. Martin’s Summer

At the risk of annoying those who might find this a bit trivial, I beg to extend the silly season for a short while longer. The rules for using numbers do not seem to be the same as the rules for using words. In using words, it is assumed that unique, individual entities are being designated. These are grammatical subjects and predicates that are given names. They appear as being separate and distinct from each other and therefore can be near neighbors without losing their identities. Numbers, however, such as rational and irrational numbers, cannot be near neighbors because they exist on a continuum. They gradually blend into each other by fine degrees, too infinitesimal to be distinguished by the human mind. Whenever there appears to be a gap between them, the gap seems to fill up with more rational and irrational numbers. Their unique individualities are seen as being smeared in a spread of continuity. Dedekind proposed that we could cut the continuity at any location and create irrational numbers to fill any gaps. If these operations with numbers and words refer to mental constructions, then the rules that were invented by humans for these game-like operations cannot be the same for both numbers and words. These conventional rules do not allow us to use nouns like neighbors, nearness, distance, or continuity when referring to numbers, which are totally different from words. Therefore, such poetic metaphors and category mistakes should not be passively tolerated in the Wikipedia article on mathematical analysis.69.19.14.37 20:34, 14 January 2007 (UTC)AubreyAubervilliers

Nice 'cut and paste' from a philosophy essay. Good work. Words like neighbourhood are perfectly valid once you understand that it itself can cover a continuum of different cases, e.g. all real numbers x such that |x| < ε; choose ε! (not ε factorial, but ε exclamation). Just as there is no one idea of a neighbourhood, there are many meanings for a given word, e.g. the noun 'light' does not have a single well defined meaning; there are different kinds of light. Light comes from the electromagnetic spectrum, members of which have different and similar properties. Just like topological neighbourhoods! But this is just a simple discussion of metamathematics, and so is at home on this comment page about an article on analysis.  Δεκλαν Δαφισ   (talk)  20:07, 13 February 2009 (UTC)

## Analysis is not a Hodge (pun intended) Podge

Okay, now I'm being a little silly! Thet being said, the openin sections of this article could easily give the reader the impression that analysis is a more or less random collection of topics (limits, continuity, derivatives and integrals, approximation, topology, etc.), providing no basic sense of unity. Why does analysis include such things as dynamical systems, measure theory, numerical analysis, and sso forth? What about more abstract ideas such as Hilbert an Banach spaces? Where does global analysis (hence the allusion to Hodge theory!) fall into all of this? I don't think I have any simple answers here. No one really seems to be able to define mathematical analysis, but everyone knows it when they see it. There are common themes of course: the most obvious of which are perhaps approximation and continuity. But thesse concepts require context, and ultimately can take you full circle. No one has mentioned heat flow, but it is a perfect example of a phenomenon that involves gradual evolution of a system based action in the small. But analysis of the heat equation leads rapidly to Fourier series, and hence to Hilbert spaces. But then we find ourselves asking the same sorts of questions we might ask in ordinary calculus -- albeit in an infinite dimensional space. And that is a core idea: We start out with elementary analysis/calculus, move on to a more complex system, but find that we are then dealing with concepts that are formally similar, but the kinds of things that vary are, well, more complex. Ordinary differential equations have to do things like the motion of a discrete body subject to a force, but when you think about heat flow, what is "moving" is not a point, but something with spatial extent. That gives you infinitely many points, and essentially just a dynamical system of the old familiar type in an infinite dimensional space (like a Banach space)! Okay -- are those meandering thoughts totally confused or what? Greg Woodhouse 22:03, 19 March 2007 (UTC)

Valid point! Would you like to write a proper lead, then? Arcfrk 04:32, 22 March 2007 (UTC)

## Hard Analysis

I don't think hard analysis should be characterized by not making use of functional analysis; for instance, the geometry of Banach spaces certainly uses these concepts but is usually characterized as "hard." I would say that soft analysis has a focus on sets, where hard analysis has an emphasis on approximation and quantity, but even this isn't an exact division...--74.132.206.251 19:44, 29 October 2007 (UTC)

I agree. The term hard analysis doesn't refer to a particular topic, but rather to the style: for instance the use of long heavy computations and inequalities. So even in a single theorem one can distinguish the soft and the hard analysis. --pma (talk) 08:33, 13 February 2009 (UTC)

## How is this different from calculus?

what distinguishs mathematical analysis from calculus in general? BriEnBest (talk) 07:28, 4 February 2008 (UTC)

The boundaries aren't hard and fast, but I would say that analysis is a wider field than calculus. You could classify real analysis and maybe complex analysis as calculus, but I would say that functional anaylsis, non-standard analysis and p-adic analysis are not calculus. As a concrete example, I would say the Baire category theorem is a theorem of anaylsis but not a theorem of calculus. Gandalf61 (talk) 10:21, 4 February 2008 (UTC)
And in a very informal sense: analysis is what calculus became when it grew up.  Δεκλαν Δαφισ   (talk)  20:04, 13 February 2009 (UTC)

Quite what does "nearness" mean in a topological sense? The article seems to say that it is a well defined property for elements of a topological space. It's clear in a metric sense, but in a topological one?! I assumed that two points x1 and x2 in a topological space X may have been said to be "near to one another" if there existed an open neighbourhood ${\displaystyle U\subseteq X}$ such that ${\displaystyle x_{i}\in U}$ for ${\displaystyle i=1,2}$. But then X is, by definition, open so all points in X are, by definition, "near to one another". So then does one ask that U is a proper subset of X? I'm a mathematician, and I can't get to the bottom of what is meant by "nearness", so does a newcomer have any hope?  Δεκλαν Δαφισ   (talk)  17:01, 13 February 2009 (UTC)