Existence of bounded linear operator with kernel reduced to $\{0\}$

Question

If $X$ and $Y$ are normed spaces, why there must exist a bounded linear operator $T$ from $X$ to $Y$ such that $T(x)$ is not equal to $0$ for all non-zero $x$?

Answer 1

This is not true. Take $X=\mathbb{R}^n$ and $Y=\mathbb{R}^m$ with $n>m$.

More generally, take $X$ and $Y$ such that density character of $Y$ strictly greater than density character of $X$.

In these cases all linear operators from $X$ to $Y$ are non-injective.