Injective compact linear operator from infinite dimensional normed space

Question

Let $V$ be an infinite dimensional normed space. How can I show that there exist a normed space $W$ and an injective compact linear operator $T$ from $V$ to $W$?

Answer 1

As I mentioned in the comments, the question has a negative answer if $V$ is too big.

To be precise, suppose that $T$ is an injective compact linear operator from $V$ to $W$. As it is well known, the range of a compact operator is always a separable space and hence $T(V)$ is separable. This in turn implies that the cardinality of $T(V)$ is at most $2^{\aleph_0}$. Consequently, if $V$ has a cardinality strictly bigger than $2^{\aleph_0}$, no compact operator defined on $V$ is injective.

On the other hand, this question is likely to be a lot more interesting if one assumes that $V$ and $W$ are separable spaces. In that case the relationship between $V$ and $W$ is bound to play a much bigger role. Observe that my reasoning above completely ignores $W$.