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The definition of subspace from the Friedberg book :

A subset $W$ of a vector space $V$ over field $F$ is called a subspace of $V$ if $W$ is a vector space over $F$ under the operations of addition and scalar multiplication defined on $V$.

say our Field is $\Re$, and let $V$ consists of vectors "$a_n(i)$"(a vector with only one dimension is considered for simple explanation ) where $a, n \in N$ , if $W$ has to be a subspace of $V$, then $W$ has to be a subset of $a_n(i)$, i.e. $ (i, 2i, 3i,...)$, now if consider $i$ and $2i$ to be forming $W$, then because of the addition property $3i$ has to be there in the set of $W$, if $3i$ is there then $4i$ has to be there in $W$ because of addition property, this goes on and we have to exhaust the original vector space $V$, so what is the subspace $W$? and also the example that I have considered here, does $V$ satisfy the addition and scalar multiplication property to be a vector space? so I have two questions:

1-Is the example considered here is a valid vector space and if yes, then

2- what can be its subspace?

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    It's really hard to understand your notation. Subscripts are completely unnecessary for describing a generic element of a vector space, unless you are taking pains to describe an element with respect to a basis.2012-08-03
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    I don't understand what your set $V$ looks like, but it sounds like it is not a vector space.2012-08-03
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    It is not clear enough to be sure, but it looks as if you are saying that a $1$-dimensional space $V$ will not have many subspaces. And that is true. A $1$-dimensional space has two subspaces: (i) the subspace consisting of the $0$-vector, and that's all and (ii) $V$ itself. But any space (over the reals) of dimension $\ge 2$ has infinitely many subspaces.2012-08-03

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