Vector space with the finite topology

Question

Let $V$ be a $\mathbb{C}$-vector space and topological space.

Suppose the topology of $V$ is the finite topology, i.e. $U \subset V$ is open if and only if $U \cap D$ is open in $D$ for each finite-dimensional linear subspace $D \subset V$, where $D$ is given the standard Euclidean topology.

Then, I hear that

$V$ is not a topological vector space if $\dim V \geq 2^{\aleph_0}$ .

Question: How do I prove this? And I want to see the reference to this fact.

Answer 1

Here are some references:

Kakutani, Shizuo, and Victor Klee. "The finite topology of a linear space." Archiv der Mathematik 14.1 (1963): 55-58

Dugundji, James. "Topology." Allyn and Bacon Inc., Boston (1966) - Appendix One 4.3

The addition is not continuous if you have a vector space with the finite topology and uncountable dimension.