Suppose $X$ and $Y$ are two infinite dimensional Banach spaces. What can we say about the set of all injective continuous linear operators between $X$ and $Y$? Is it always nonempty? 



Injective continuous operators between Banach spaces
Of course the dimension is an obvious obstacle, but even if the space have the same cardinality of Hamel bases the answer is no. For example in the paper


**A. Avilés, P. Koszmider,** 
A Banach space in which every injective operator is surjective.
*Bull. Lond. Math. Soc.* 45 (2013), no. 5, 1065–1074


the authors constructed an infinitely dimensional Banach space $X$ such that if $T:X\to X$ is bounded and injective, then $T(X)=X$. Therefore if $Y$ is a subspace of $X$, then one cannot find an injective operator $T:X\to Y$.


