Difference between 'Field' (algebra) and 'Field' (geometry)

[FluidStu]

I am trying to build up a kind of mind map of the following:

Module (eg. vector space)
Ring (eg Field)

Linear algebra (concerning vectors and vector spaces, from what I understood)
Multilinear Algebra (analogously concerning tensors and multi-linear maps)
Linear maps & Multilinear maps

The first question I have is, what is the difference between an algebraic field and a geometric field? I understand what scalar, vector and tensor fields are – are these examples of geometric fields? If so, can they be considered as 'Rings', or are these geometric fields totally separate from the algebraic ones?

The second question is about spaces. I'm trying to build a mind map of these (using the image here as a basis (https://en.wikipedia.org/wiki/Space_(mathematics)#Types_of_spaces)), and relate them to the above. Are vector spaces 'spaces'? By that I mean, do they come under the classification of a mathematical space, as defined in the Wiki link?

Many Thanks

 

[andrewkirk]

I would use the term topological field, rather than geometric field, as topology is a more general superset containing geometry. A topological field is a function from a topological manifold to a vector space, including the cases of scalars and tensors as vector spaces since they can both be characterised as vectors.

A field in algebra bears no relation to this. It is, as you say, 'totally separate'. The use of the same word is - so far as I know - simply coincidental - a bit like how the word 'normal' has many different meanings in different contexts in mathematics. A field in algebra is a set ##S## of elements on which two binary functions (from ##S\times S## to ##S##) are defined. The functions are called 'addition' and 'multiplication' and must obey a certain set of axioms.

Are vector spaces 'spaces'? By that I mean, do they come under the classification of a mathematical space, as defined in the Wiki link?

Vector spaces intersect with the hierarchy shown in your link, but neither is contained within the other. The hierarchy in your link is of topological spaces. Normed vector spaces have a topology, and thus are shown in your hierarchy. But not all vector spaces have norms. A vector space without a norm is a purely algebraic object, as it doesn't generally have a topology (although there may be some cases where a topology can be imposed without introducing a norm), and the defining axioms for a vector space are purely algebraic.

The word 'space' generally refers to a topological object. Non-topological vector spaces are the only common exception I can think of to that.

 

[Hornbein]

Physicists and mathematicians assign different meanings to the word "field." There is no relation.

 

[FluidStu]

I would use the term topological field, rather than geometric field, as topology is a more general superset containing geometry. A topological field is a function from a topological manifold to a vector space, including the cases of scalars and tensors as vector spaces since they can both be characterised as vectors.

A field in algebra bears no relation to this. It is, as you say, 'totally separate'. The use of the same word is - so far as I know - simply coincidental - a bit like how the word 'normal' has many different meanings in different contexts in mathematics. A field in algebra is a set ##S## of elements on which two binary functions (from ##S\times S## to ##S##) are defined. The functions are called 'addition' and 'multiplication' and must obey a certain set of axioms.
Vector spaces intersect with the hierarchy shown in your link, but neither is contained within the other. The hierarchy in your link is of topological spaces. Normed vector spaces have a topology, and thus are shown in your hierarchy. But not all vector spaces have norms. A vector space without a norm is a purely algebraic object, as it doesn't generally have a topology (although there may be some cases where a topology can be imposed without introducing a norm), and the defining axioms for a vector space are purely algebraic.

The word 'space' generally refers to a topological object. Non-topological vector spaces are the only common exception I can think of to that.

Thanks. So when we talk about a vector space (or generally a module) being 'over' a field (or generally a ring), we are referring to an algebraic field? And we are not referring to a general 'Space' (as defined in link)? (since these 'spaces' are topological in nature, and therefore can't really be related to the algerbraic 'field'?)

So vectors spaces can be topological (part of the 'Spaces' wiki page), but are not always? (having a hard time with getting 'algebraic' and 'topological' to be included in the same big mind map.) To add more confusion (for me), vector spaces are also called linear spaces – but there is also a separate 'linear space (geometry)' page on Wiki. Mind in meltdown.

Finally, at the risk of stretching this out beyond the topic title, I would like to tie in the "linear algebra" and "multi-linear algebra" to the idea of the vector space. My understanding so far is that, Linear Algebra transforms one vector space (aka linear space, i.e collection of vectors) to another through a 'linear map'. Bi-linear algebra takes two vector spaces and uses a 'multi-linear map' to give a third vector space. An example would be matrix multiplication. (Bi-linear algebra and maps just being the first examples of multi-linear algebra/maps).

 

[andrewkirk]

Thanks. So when we talk about a vector space (or generally a module) being 'over' a field (or generally a ring), we are referring to an algebraic field? And we are not referring to a general 'Space' (as defined in link)? (since these 'spaces' are topological in nature, and therefore can't really be related to the algerbraic 'field'?)

Yes, that's right.

So vectors spaces can be topological (part of the 'Spaces' wiki page), but are not always?

Yes. A vector space with a norm - which is a measure of the 'size' of a vector that obeys certain rules - can be given a topology, which imparts a sense of 'connectedness' and 'closeness' on different members of the space. Without a norm, there is no general way to do that. The usual norm for the simplest vector spaces - copies of ##\mathbb{R}^n## - is ##\|\vec x\|\equiv\sqrt{\sum_{k=1}^n x_k{}^2}##.

An example of a vector space that has no norm (I think. I'm happy to be corrected on that) is the set of all functions from [0,1] to ##\mathbb{R}##. This is an infinite-dimensional vector space. Norms can be put on some vector spaces of functions but they require additional constraints, such as continuity.

Linear Algebra transforms one vector space (aka linear space, i.e collection of vectors) to another through a 'linear map'. Bi-linear algebra takes two vector spaces and uses a 'multi-linear map' to give a third vector space. An example would be matrix multiplication. (Bi-linear algebra and maps just being the first examples of multi-linear algebra/maps).

That's a Linear Map, not Linear Algebra. Linear algebra is the mathematical discipline that concerns itself with linear maps. The study of multilinear maps - which includes bilinear maps as a special case - is included in the broad discipline of Linear Algebra. A multi-linear map is a map that takes as input one vector from each of a number of vector spaces, and gives a result that is a vector in another space, and the map is linear in each of its arguments. Some or all of the vector spaces may be the same as each other.

 

[Samy_A]

An example of a vector space that has no norm (I think. I'm happy to be corrected on that) is the set of all functions from [0,1] to ##\mathbb{R}##. This is an infinite-dimensional vector space. Norms can be put on some vector spaces of functions but they require additional constraints, such as continuity.

Small nitpick: if you accept the axiom of choice, any vector space can be given a norm.

But you are correct in that many interesting topological vector spaces have a topology that can not be derived from a norm.

 

[FluidStu]

Ok, little bit more mind mapping and reading. (Background to question: Tensors! Trying to get my head around them, and so would like to build a picture of how they connect with other mathematical 'things'). I've tried to define what my understanding is at present, with questions in blue:

A vector space lies 'over' an algebraic field.

Vector space = collection of vectors which may be added together or multiplied by scalars. (I'm assuming that the italic part is the difference between a vector and a vector space?). These scalars may be real numbers, complex numbers or generally any FIELD–>

note: vector spaces come under the 'module-like' category of algebraic structures
note: normed vector spaces are a kind of topological 'space'. This is shown by the green part of the image shown at the top of this page: (https://en.wikipedia.org/wiki/Space_(mathematics)).​

Field = an algebraic field is a set (e.g. all real numbers) comprising many elements (eg the number 2 and number 9 etc) for which the rules of addition and multiplication are defined by axioms (rules).

note: fields come under the 'ring-like' category of algebraic structures​

Topological Fields are different. They assign a scalar, vector or tensor to each point in a mathematical space (making a scalar, vector or tensor field respectively). (This space is geometric, right? Like Euclidean space?)

Linear algebra is concerned with transforming a vector space (or generally module?) into another vector space through a linear map (linear function).
Bi-linear algebra is concerned with transforming two vector spaces (or generally module?) into one other vector space through a bi-linear map (bi-linear function). An example would be matrix multiplication. The resulting vector space is linearly related to both parent vector spaces. (https://www.quora.com/What-is-multilinear-algebra-Is-it-a-generalization-of-linear-algebra )
Multi-linear algebra = ? (It would be good to have a definition analogous to those for linear and bi-linear algebra)

Tensors are geometric, and describe linear maps between given scalars, vectors and tensors. (Wiki) (i) (So the tensor is the 'description'? This is confusing) (ii) (Why only linear maps?). Wikipedia gives three ways to define them:

>> When given a basis (e.g. a co-ordinate system), they can be represented as a multi-dimensional array.
>> They can be defined as a Multi-linear map. (i) (How? Isn't this in disagreement with the above point about tensors being linear maps?) (ii) Does the answer to (i) it have something to do with the tensor product? Is this what was meant by "However, it's possible to create a space Z such that linear maps f : Z -> W correspond to bilinear maps f : U x V -> W. This space is called the tensor product of U and V, denoted U⊗V"? See here: https://www.quora.com/What-is-multilinear-algebra-Is-it-a-generalization-of-linear-algebra ) which brings me to the last definition...
>> Using tensor products.​

 

[andrewkirk]

Small nitpick: if you accept the axiom of choice, any vector space can be given a norm.

Interesting! I didn't know that, although I suspected there might be a fact like that lurking out there. I suppose I'm not surprised the Axiom of Choice can achieve such a thing. After the Banach-Tarski paradox, it's hard to be surprised at any counter-intuitive result being derived from AC.

 

[zinq]

"Physicists and mathematicians assign different meanings to the word "field." There is no relation."

Not true. Mathematics is full of algebraic fields like the real numbers, the complex numbers, fields of rational functions (quotients of polynomials with coefficients in some other choice of field like the reals) and finite fields (there is exactly one of these, of order pⁿ for each prime number p and positive integer n).

But also, mathematics is full of geometric fields like scalar fields, vector fields, and tensor fields.

So mathematicians, for better or worse, use the word "field" in both senses, which are distinct meanings, All. The. Time.

 

[micromass]

Interesting! I didn't know that, although I suspected there might be a fact like that lurking out there. I suppose I'm not surprised the Axiom of Choice can achieve such a thing. After the Banach-Tarski paradox, it's hard to be surprised at any counter-intuitive result being derived from AC.

Including counter-intuitive results like "Every infinite set contains a countable set"

 

[Erland]

It is unfortunate that one in English has chosen the same word "field" for vector and tensor fields, on the one hand, and algebraic fields on the other hand, because these are completely different things. In German, the latter is called "Körper" (in Swedish: "kropp", which corresponds to the German Word), which means "body". I think it would be better to use the word "body" for an algebraic field in English, too, but it is probably too hard to accomplish a change...

 

[zinq]

Yes, but it's not a real problem. I have never heard of any case where the meaning of the word "field" in mathematics was not completely obvious from context.

 

[mathwonk]

vector and tensor fields actually give an important example of modules, since you can add elements of these topological-geometric "fields", and you can also multiply each vector or tensor by a number, so you can multiply the whole vector-tensor field by a function. so the set of tensor fielods of a certain type is a module over the ring of functions. inded vector fields correspond to wha are called "projective" modules, or locally free modules, since tangent and tensor bundles are locally trivial.