Is a surjective linear transformation from an infinite dimensional vector space $X$ to $X$ always a one-one mapping?

Question

Is a surjective linear transformation from an infinite dimensional vector space $X$ to $X$ always a one-one mapping?

(In case of finite dimensional, surjectivity implies one-one. What about infinite dimensional vector space?)

Answer 1

No. The derivative on the space of polynomials is surjective but not injective.

Answer 2

The standard counterexample to this is the shift-left map, on the space of infinite sequences of numbers:

$$ T(x_1,x_2,x_3,\ldots) = (x_2,x_3,x_4,\ldots) $$

Answer 3

This has been already answered but let me remark that there exist spaces where this is actually true as long as by a linear map the OP means a continuous linear map on a normed space.

Then all hereditarily indecomposable Banach spaces have the property that surjectivity implies injectivity (all Fredholm operators acting on such spaces have index zero, hence surjective implies injective in this case).