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Assume $V$ and $W$ are isomorphic infinite dimensional vector space over a field $k$, and $f:V\to W$ is an injective $k$-linear map. Is $f$ an isomorphism of vector spaces? If so, does category theory generalize this statement? That would be a theorem such as "all monomorphisms of isomorphic objects are isomorphisms."

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up vote 3 down vote accepted

Multiplication by any non-constant polynomial $P$ is a $K$-linear map $K[X]\to K[X]$ that is injective (since $K[X]$ is an integral domain) but it is never surjective (since $P$ is not invertible, the constant $1$ is not in the image). So clearly what you suspected is not true in general.

More generally the failure of what you say allows for the existence of rings containing a field $K$ that are integral domains without being fields, and such rings abound. Your statement does hold true for finite dimensional vector spaces, so one can conclude that the mentioned type of ring can never be finite dimensional over its subfield$~K$. (And by a similar argument, without the requirement of containing a field, an integral domain that is not itself a field cannot be finite.)

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The statement is not correct. Consider $V=\mathbb{R}^\mathbb{N}$, $f:V\rightarrow V$,

$$f((x_1,x_2,\dots))=(x_1,0,x_2,0,x_3,0,\dots)$$

This is injective, linear and not an isomorphism.

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You surely must mean $V=l^p$ or some other infinite dimensional space. –  Student Feb 4 at 12:37
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@Student: $\mathbb{R}^\mathbb{N}$ is the set of all sequences of real numbers. This is an infinite dimensional vector space w.r.t. componentwise addition and scalar multiplication. –  Your Ad Here Feb 4 at 12:49
    
I see what you mean, thank you. –  Student Feb 4 at 13:07
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