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First of all, I would like to ask to you, to restrict the answers just in the context of Real Vector Spaces. So, my question is quite simple: what supposed to mean the phrase: "this map preserves the vector space structure. Or, under these defined operations, the vector space structure is preserved". For instance:

Let F be a field. And V and W vector spaces.

Consider,now, a map:

$T: U \to V$; $u_{1}\boxplus_{U} u_{2} \mapsto T(u_1\boxplus_{U} u_2) := T(u_1)\boxplus_{V} T(u_2) $

Well, what is the difference between the $\boxplus _{U}$ and $\boxplus _{V}$? And what this difference is supposed to mean an how is related with the phrase? What is the undoubtly right way to interpret these kind of notions everytime?

[My first guess is the thought: "well, under the $\boxplus _{U}$ the $T(u_1\boxplus_{U} u_2)$ is an element of a set V. If this V forms a vector space under the $\boxplus _{V}$ (why occurs the change of $\boxplus _{U}$ and $\boxplus _{V}$?) ,then this is the meaning of "preserve structure".]

Obs:------------------------------

Is there any sense to ask the difference of: A map that preserve structures and a operation that preserve structures?

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  • $\begingroup$ If you want us “to restrict the answers just in the context of Real Vector Spaces”, then did you tag your question under general-topology? What has this to do with it? $\endgroup$ – José Carlos Santos Sep 9 '17 at 10:44
  • $\begingroup$ @JoséCarlosSantos. You're welcome to edit out the general topology tag. $\endgroup$ – William Elliot Sep 9 '17 at 10:53
  • $\begingroup$ @WilliamElliot I know, but I would like to know wht the OP used it in the first place. $\endgroup$ – José Carlos Santos Sep 9 '17 at 10:54
  • $\begingroup$ And I'm wonder if my question is too hard and I know that it's not, I mean: it is cleary problems with notation and the true meaning of "Preserve structure". $\endgroup$ – M.N.Raia Sep 9 '17 at 11:00
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    $\begingroup$ @Jack Clerk, that's not how it works. You are more likely to annoy those that are looking for questions in topology than attract them. And everyone else for clearly abusing the site mechanics. $\endgroup$ – Ennar Sep 9 '17 at 11:06
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A vector space $V$ over a field $F$ is a set together with certain operations such that certain conditions are fulfilled. In particular, we have a binary operation $V\times V\to V$ that we call addition and might denote as $\boxplus_V$ and that makes $V$ an abelian group, and we have an action of the field $F$ on this group, i.e., a map $F\times V\to V$ (which we might want to write $\boxdot_V$). If $U$ is another vector space, we have the same kind of structure, but use different symbols $\boxplus_U$, $\boxdot_U$ to denote them. Of course, $\boxplus_U$ and $\boxplus _V$ are different maps, already because they are defined on different domains, similar for $\boxdot_U$ and $\boxdot_V$. (Actually, we do usually use the common symbols $+$ and $\cdot$ throughout and hope that no confusion arises, but to cover the most general situation, e.g., $V$ is already endowed with an addition, but we want to us a different operation, or $U$ and $V$ have elements in common, but the sums of these differ, we better be careful with the notation)

So if we pick three vectors $u,v,w\in V$, it may (or may not) be the case that $w$ is the sum of $u$ and $v$. A structure-preserving map $T\colon U\to V$ now is one that preserves the sum relation, i.e., whenever we have $u,v,w$ such that $w$ is the sum of $u$ and $w$ ($w=u \boxplus_V v$, then their images $Tu, Tv, Tw$ are in the same (or rather corresponding) relation ($Tw=Tu\boxplus_U Tv$). Of course, we also postulate that $T$ respect the scalar multiplication, i.e., for any $\alpha\in F$. whenever $v,w\in V$ are in the relation that $w$ is thw $\alpha$-multiple of $v$ ($w=\alpha\boxdot_V v$), then so are their images$Tv$ and $Tw$ (i.e.,$Tw=\alpha\boxdot_U Tv$)

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The simple answer is that what is meant is $T$ is a linear transformation from $U$ to $V$.

In other words:

$T((a\boxdot_U u_1) \boxplus_U (b\boxdot_U u_2)) = (a\boxdot_V T(u_1)) \boxplus_V (b\boxdot_V T(u_2)).$

Usually the operations are understood to be the operations that make $U,V$ vector spaces in the first place, and we write simply:

$T(au_1 + bu_2) = aT(u_1) + bT(u_2)$.

Put another way, linear combinations (over $F$) of elements of $U$ are mapped by $T$ to the same linear combination of the images (under $T$, over $F$).

So the "vector-space-ness" of $U$ is preserved in the image (range) of $T$ in $V$.

One has to be careful, here: it is usually not the case that we have a "copy" of $U$ inside $V$, because $T$ may not be one-to-one. What typically happens is that $T$ condenses the space $U$ into a (perhaps) smaller subspace of $V$.

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