It looks like a statement, though: "Topological vectors". "Vectored topology"...? Topology is to do with geometry, isn't it? I thought topology was to do with surfaces and "spaces". E.g. the topology of the e^st surface is a circle, and the domain is complex frequency. Vectors have an amplitude or a size, and a direction, or an angle. Where does geometry disconnect from this "model"?
Where did I say otherwise? There's a difference between rigorous mathematics and applied mathematics. Applied mathematics might be using calculus of variations to compute the motion of a spinning top. Rigorous mathematics is justifying that your notion of dynamical systems is valid and that your Lagrangian density for the top exists within the relevent Schwartz space (ie your action is well defined). Might sound needlessly picky but more than once physicists have been caught out because they overlooked something buried deep in the mathematics, particularly in the last 100 years. No, I mean you shouldn't assume that the proofs or derivations or methods of explaination of mathematical results found in a book pitched at non-mathematicians is going to be the way mathematicians would go about it. Engineers are more practical. They don't care how you go about justifying the inverse of a Laplace transform, only that there's a formula (or a vast collection of standard inversions) for them to use. Does it matter to an engineer that you cannot indiscriminantly swap the order of integration when proving the convolution formula for a Laplace transform? I doubt it, just that there's a convolution formula. The complex plane is one diagrammatic representation of the set 'The Complex Numbers'. There are many ways to represent it, the plane is the more straight forward because most children are familiar with the notion of 'The number line'. The entirity of the plane can be represented by a sphere, a disk or just about any shape you want, provided you're careful with the details. You can connect any set of numbers to geometry if you want, provided you're careful with the details. You have to turn the set into something like a normed space or a metric space. For instance, it's somewhat second nature to think "The distance between 3 and 4 is 1" because 4-3=1. They are '1' apart. But that's only one way of looking at it. The notion of distance (distances are always positive, except the distance from a point to itself, which is zero and also the distance from x to y and then from y to z totals the same or more than the distance from x to z) allows for infinitely many geometric constructions of the same set of entities. The example I just gave for the Reals uses : d(x,y) = |x-y| > 0 unless x=y But I could have something like D(x,y) = 1 if x != y and D(x,y)=0 if x=y. That's entirely consistent and valid in the complex 'plane' too. Thus it's important to seperate the notion of a set of entities and some possible ways of arranging them and inter-relating them to one another. All the stuff about trig and hyperbolic functions you mention have nothing immediate to do with geometry if you look at them purely in terms of functions. They do have applications but you can study functions like \(s(x) = \sum \frac{(-1)^{n+1}x^{2n+1}}{(2n+1)!}\) purely to see what kinds of properties they have. That's how my analysis teacher did it. Riemannian manifolds are a subclass of manifolds (they are differentiable). Manifolds are a subclass of topological spaces. Topological spaces are particular kinds of sets with additional structure. All in all, a Riemannian manifold is not a general space in terms of what other things exist in mathematics which involve infinitely many points/elements with structure. They just come up a lot in physics because we want a notion of 'somewhere', 'some time' and smooth changes. Smooth manifolds provide just that. What is the purpose of this forum? Why do people come here? To discuss things. I assume some people come to learn, either a lot or a little. As such, if someone knows another poster is in error and can offer comments, that furfills the purpose of this site. When it comes to mathematics, opinion often matters little. 2+2=4, wether I like it or not. There doesn't exist a globally definable gauge potential on a 2-sphere, wether Dirac liked it or not (you cannot comb the hair of a bowling ball). I speak from experience that it's better to just pipe up and say "I'll admit, I haven't a clue". I've said that to supervisors and professors and always benefited from saying it because they reapproach the problem rather than continuing to go over your head. There's nothing wrong with saying the wrong thing, provided it's not a deliberate attempt to appear like you know the right thing. I can point at a few members here I know from other forums who have absolutely no problem claiming to be competant at quantum field theory, string theory or group theory but when they post about such things, their lies are transparant. At some point or other, we all were ignorant about any given section of physics and I know I've said plenty of things which turned out to be completely wrong later on. It was to my benefit I didn't kid myself into thinking I was right and ignored ways of learning which would correct me. Case in point. You seem to suddenly shift your discussion entirely and pounce on a topic, pulling questions out which give the impression you've just stumbled across a bit of maths you've no idea about and want to dive on in head first. 'Regular' can have many different meanings, depending on what specifically you're referring to. That's why you might feel you see contradicting definitions. A manifold to an engineer might be a method of pumping a fluid through pipes as well as a topological space with a series of particular charts defined on it. A 'regular polygon' is one whose sides and angles are all the time. That would be different from a 'regular space], a completely regular space and a regular graph, but they all seem to be in the same general field but still the distinction is important. In much less geometric areas you can get things like regular rings (though via algebraic geometry you can still give such things geometric considerations). Is there a particular site/book/paper which prompted your question on regularity?
No particular paper or book; as you surmise, it's something I "came across" (and I recalled having encountered it before), so I wanted to "explore" what it's supposed to mean. The earlier questions about compactification etc, are because I've been trying to follow the discussions and blogs that have sprung up over a certain just-characterised group: E8, and "embedding". This is adv. group theory, though, so it's a bit like trying to follow a tricky surgical procedure or something, when you haven't studied medicine. P.S. Any impressions are more likely due to my style, which tends to be "discontinuous", shall we say. The "shift", you seem to have perceived is due (when I look again at my post), to my "reaching the end", and so, posing a question. Otherwise it looks more like just a whine, with bugger-all point to it, IMO. But, note the new "packaging" I'm using. P.P.S. I think about terminology a bit (constantly, perhaps), and a word like "regular", obviously has various shades of meaning - the mathematical meaning is supposed to be precise, and I'd like to understand more about how it applies to spaces, general or otherwise. In sound (music), there's a lot of regular and irregular harmony, which is intervals and ratios. Currently I'm into the Romantics (the fathers of modern Jazz), which (I think) explores the regular/irregular relations of harmonics. But that's really just coincidental.
The meanings are precise, it's just that there might be more than one definition depending on what area you are working in. Ask a geometry or GR person what 'ring' means to them and they'll probably describe something like a torus. As someone who does algebraic stuff what a ring means to them and you get back this. Both are correct. There's only so many 'nice words' mathematicians end up using. You can call new structures or entities anything you like but often words like 'regular' or 'simple' are convenient because even qualitatively they convey a little bit of understanding. Unfortunately, they are also already in use, but provided it's clear from the context which 'regular' or which 'simple' you are using, there's never any problem to someone who knows both meanings. For instance, if I were talking to someone doing logic and axiomatic arithmetic systems, their use of 'simple' would probably be in reference to how people like Godel defined the complexity of systems. If I'm talking to someone about Lie groups, Lie algebras and their Killing forms, 'simple' will refer to the group theory notion of a Lie algebra with no trivial ideals. The context makes it obvious. I don't expect to have to reach for a Latin dictionary to translate a Japanese newspaper, just as in all probability, a group theorist used group theory terminology.
I said "Topological vectors" like we do when someone says something we don't understand and we repeat it as a way of asking for clarification. I'd say that's actually a relatively recent fad, historically speaking; from that vantage, the "difference" is certainly artificial.
Just want to mention that in Chapter 4 of Rudin's Functional Analysis he defines adjoint the way you define transpose here, with different V and W. But everywhere else seems to define adjoint for V=W and even sometimes only when V=W is Hilbert.
Not that recent. A History of Mathematics, Florian Cajori, which covers the history of mathematics from antiquity to 1919 has a chapter devoted to "Applied Mathematics". Why only 1919? Because that is the year the book was published. (Unfortunately the posted link ends around page 73. Applied mathematics is 370 pages later.) That said, applied mathematics as performed by mathematicians is not the same thing as the pseudo-mathematics used by many engineers and even some scientists. Applied mathematicians are still quite rigorous. Kolmogorov might well come back from the grave and haunt you if you insist on saying otherwise.
He already haunts me with his intractable second year analysis book (translated into English) which more or less assumes you already have a PhD. :bawl:
I meant the distinction between applied math and pure math. Newton for example, never talked about any such difference. I don't even know if he recognized such a difference. People like Hardy and Bourbaki really sought to make mathematics for the sake of mathematics ie. abstraction.
Well, what do you mean geometric? There are geometric concepts which are involved in linear algebra, like angle between vectors and so on. But in linear algebra, a function can be a vector. A matrix can also be a vector. Even a number can be a vector, contrary to what we were taught earlier in school. It's hard to think geometrically about what the angle of a matrix or a polynomial means. It just happens that from physics/calculus we only think of 3d vectors. The basic reason is that we dealt with different definitions of vectors in physics/calculus than is dealt with in college linear algebra. That's why I said vectors "today". Maybe you could think that the whole reason behind linear algebra is to eliminate the need to think (but not necessarily the usefulness of thinking) geometrically. I don't think algebra as taught in college deals much with vectors, except in a linear algebra framework. It really depends on who you ask and how they define vectors. I guess you have the physics conception of vectors in mind. That conception is consistent with the mathematical conception. I just wanted to mention that in linear algebra there are more kinds of vectors than those mentioned in physics.
"...vector spaces with certain additional structures. ... for recovering ordinary notions from geometry. * A real or complex vector space with a well-defined concept of length, i.e., a norm, is called a normed vector space. * A normed vector space with the additional well-defined concept of angle is called an inner product space. * A vector space with a topology compatible with the operations — such that addition and scalar multiplication are continuous maps — is called a topological vector space. * A vector space with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field. * An ordered vector space." --wikipedia.org Hmm...The algebraic notion of vectors as numbers/structures is more like the idea of connection (which implies 2 things that get connected), and the idea of (a) projection. P.S. Interestingly the "google product" for "vector mathematic" was 4.22 x 10^6. but for "vector math" 7.32 x 10^4. I recall that a matrix has row and column vectors.
* A normed vector space which is complete with respect to the metric induced by the norm is a Banach space * If Banach space norm is induced by an inner product, it is a Hilbert space
A normed vector space is any vector space endowed with the concept of a norm, or "length" in the vernacular. Very abstract vector spaces can be endowed with a norm. Function space, for example. You do not need angle to define an inner product. An inner product is a positive-definite nondegenerate sesquilinear form on a field. What is the angle between two functions? An inner product can be used to define a norm. The converse is not true. The taxicab norm does not arise from a inner product.
An angle connects the idea of a projection/vector back to geometry (and dot products of vector algebra). Which is where the first line of my PP points to - the land of geometry and angles (phase and rotation).
Look, in spite of occasional appearances to the contrary, mathematicians are no less human than the rest of us - they frequently resort to visual metaphors to aid their thinking about abstract concepts. The difference between them and us is that they realize they are just that - metaphors. So when you are told that a vector has "length" and "direction", you are meant to accept it in that sense, and in that sense only - you are being given the gift of a metaphor. And if you repay that gift with statements like the above, you can expect to have it withdrawn. So. A vector is an abstract object with the property that it can be commutatively added to another such object to yield a third such object, each of which can scaled by elements from some field. The inner product of two vectors is a symmetric, field-valued bilinear form if the field is real, it is conjugate-linear in one argument (i.e. sesquilinear) if the field is complex. The norm of a vector is a form of this sort where the pair referred to is the identity pair. Length, angle? Where did I mention them?
Are you implying that "vector" is a mathematical term which is not geometrical? If that's your implication, then I imagine there are some physicists who would have something to say, because vectors get used as geometrical objects all the time. This is a known. The distinction between geometrical and algebraic vectors is moot; both versions are valid, and both are vectors. If a geometrical vector is a metaphor, then so is an algebraic vector. Usually, you get told about vectors with a size (length, amplitude), and a direction (angle). It's what most people who believe they have learned about vectors, will tell you a vector is. Is there an algebraic version of an angle or direction? Is geometry some kind of unnecessary add-on; algebra is all you need?
