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Linear spaces vs. vector spaces 
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#1
Feb613, 04:18 AM

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Does a vector in an absract vector space (without any further structure i.e. no inner product or norm) have the properties usually associated with vectors, that is, magnitude and direction? If not, isn't the name vector space a bit misleading and it would be more appropriate to call it a linear space?



#2
Feb613, 04:51 AM

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Hi TrickyDicky!
we can give it the usual norm, or we can give it the norm √(x^{2} + 4y^{2}) … either way, it's a vector space, but one is squashed relative to the other, and so distances and directions are changed (and equal distances or angles become unequal)



#3
Feb613, 05:16 AM

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I'm referring to the case without any norm or inner product. I agree one can compare the collinear elements of the space and say that one is twice or half as long as the other, but that doesn't allow us to assign them a specific length and therefore measure any distance. With respect to parallelism, I gathered from another thread in the relativity forum that unless you add a connection (so you'd need more structure) you cannot properly talk about parallelism, between elements of the space, at most you can talk about collinearity. So certainly you cannot assign an angle to the elements of the space either, don't you agree? 


#4
Feb613, 05:25 AM

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Linear spaces vs. vector spaces



#5
Feb613, 05:30 AM

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$$\cos\theta(x,y)=\frac{\langle x,y\rangle}{\x\\y\}.$$ By the way, some people do call vector spaces "linear spaces" or even "linear vector spaces". 


#6
Feb613, 05:46 AM

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#7
Feb613, 07:02 AM

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Of course defining vectors over the reals simply as ordered ntuples of n real numbers should suffice to avoid misunderstandings.
I guess the problem is the representation we make of them, which is usually a geometric interpretation that may lead to ambiguities as to how much structure is added tacitly to the basic abstract vector space. 


#8
Feb613, 07:26 AM

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AFAIK "vector space" and "linear space" are synonyms, in this context. However the name "linear space" is also used for a different mathmatical structure: http://en.wikipedia.org/wiki/Linear_space_(geometry)
I think the motivation for changing the name "vector space" to "linear space" is that the elements of a "vector space" can be mathematical objects which don't have any obvious geometrical interpretation as "vectors". For example the elements of a linear space might be "matrices whose elements are functions", not "numbers". I don't think the issue of norms is relevant. As Fredrik said, you can define an infinite number of different norms on any linear space. Interpreting one particular norm as a "length" in Euclidean geometry is certainly interesting and useful in physics or engineering, but it does't have much mathematical significance. 


#9
Feb713, 02:43 AM

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In #6 where I wrote " a vector space over the reals is automatically a manifold" it should say "a topological vector space over the reals is automatically a manifold".



#10
Feb713, 02:49 AM

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What is true is that there is a very canonical way to make a finitedimensional real vector space into a manifold. That doesn't mean that every topology will make the vector space into a manifold. 


#11
Feb713, 03:03 AM

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#12
Feb713, 03:14 AM

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[tex]V\rightarrow \mathbb{R}^n:x^i E_i\rightarrow (x^1,...., x^n)[/tex] This induces a maximal atlas which defines a smooth structure on the vector space V. Furthermore, the maximal atlas turns out to be independent of the choice of the basis. What you said, is that every topological vector space is a manifold. This is wrong, as the trivial topology shows. 


#13
Feb713, 04:12 AM

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Thanks for your patience. 


#14
Feb713, 04:24 AM

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If you speak about a topological vector space, then you have some topology. Saying that a topological vector space is a manifold means exactly that the topology on your space satisfies the properties. Like I pointed out, this is not necessarily true. What is true is that every finitedimensional real vector space induces naturally a topology which makes it into a manifold. There are many topologies which you can put on a vector space and which make it into a topological vector space. But there is only one topology which make it into a manifold. 


#15
Feb713, 04:33 AM

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"The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals." Is it not a topological vector space over the reals also a topological vector space over the reals locally? In that case it seems to me being a topological vector space over the reals is enough to count as a manifold since such object is trivially locally homeomorphic to a topological vector space over the reals. Sorry about the annoying repetition of phrases. 


#16
Feb713, 04:37 AM

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#17
Feb713, 04:39 AM

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#18
Feb713, 04:43 AM

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