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## Main Question or Discussion Point

Is an injective endomorfism automatically bijective?

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Is an injective endomorfism automatically bijective?

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Thanks for confirming that intuition and for the pointer to Fredholm alternative, never heard of it.finite-dimensional spaces, then yes. This is known as the Fredholm alternative: http://en.wikipedia.org/wiki/Fredholm_alternative

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WWGD

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Doesn't this follow from rank-nullity theorem?

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mathwonk

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Yes. But the Fredholm alternative also hold in some infinite-dimensional cases. There, you can't use rank-nullity.Doesn't this follow from rank-nullity theorem?

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