# How to show that two vector spaces $V$ and $W$ are the same

How to show that two vector spaces $V$ and $W$ are the same, if we know $\dim V = \dim W$ and $V$ is a subspace of $W$ ? Would it suffice to show there exists an isomorphism between them ? Any help would be much appreciated.

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You should call the vector spaces something other than $n$ and $m$, if you wish to use those symbols for the dimensions. –  David Mitra Feb 7 '12 at 18:57
Not true if they are infinite dimensional. And, as David said, don't use $m$ and $n$ for vector spaces, use capital letters, often $V$ and $W$ are used. –  Thomas Andrews Feb 7 '12 at 19:02
Thanks guys would keep that in mind for future questions. –  Hardy Feb 7 '12 at 19:07

Assuming the dimensions are finite, show that a basis of $V$ is a basis of $W$.
@Hardy Do not worry about a basis of $W$. Start with a basis ${\frak B}$ of $V$. Then show it is a basis of $W$. Once you've done this, you'll know $V=\text{span}{\frak B}=W$. –  David Mitra Feb 7 '12 at 19:29