If we have some finite dimensional vector space V and linear transformation $F: V\to V$, and we know that F is injective, we immediately know that it is also bijective (same goes if we know that F is surjective).
I'm curious if a same rule applies if V is infinite dimensional vector space, and we know that F is injective/surjective, does it again immediately imply that F is also bijective (intuitively I think it does)?
[Math] Must an injective or surjective map in an infinite dimensional vector space be a bijection
examples-counterexampleslinear algebralinear-transformationsvector-spaces
Best Answer
No, it does not. Consider the space $\ell_\infty$ of bounded sequences of real numbers. The map
$$S:\ell_\infty\to\ell_\infty:\langle x_0,x_1,x_2,\ldots\rangle\mapsto\langle 0,x_0,x_1,x_2,\ldots\rangle$$
that shifts each sequence one term to the right and adds a leading $0$ term is linear, injective, and clearly not surjective.