A topological vector space is a vector space endowed with a topology such that the sum of vectors and the multiplication by scalars are continuous.
The weight of the topology $\tau$ is the smallest cardinal $\alpha$ such that there is a basis $\mathcal B$ of cardinality $\alpha$ for $\tau$.
By algebraical dimension, I mean the classical cardinality of its Hamel basis. So, my question is: is there some infinite dimensional topological vector space $V$ such that the weight of $V$ is larger than the dimension of $V$?
If useful, I'm okay with using some axioms such as the Continuum Hypothesis. To give some context, my interest in this question arose from a theorem that holds if the weight of the topology is less than or equal to the dimension. So, I want to know if there are counterexamples if the weight is greater than the dimension, which includes knowing whether there are spaces with this property.
Consider $\mathbb{R}^\infty$, a countable-dimensional vector space with the colimit topology with respect to its finite-dimensional subspaces (that is, a subset is open iff its intersection with each finite-dimensional subspace is open). I claim that $\mathbb{R}^\infty$ is not first-countable and thus has uncountable weight.
To prove this, let $(e_n)$ be a basis for $\mathbb{R}^\infty$ and suppose $(U_n)$ is any countable family of neighborhoods of $0$. For each $n\in\mathbb{N}$, choose $r_n\neq0$ such that $r_n e_n\in U_n$ (such an $r_n$ must exist since $U$ is a neighborhood of $0$), and let $C=\{r_ne_n:n\in\mathbb{N}\}.$ Any finite-dimensional subspace of $\mathbb{R}^\infty$ contains only finitely many points of $C$, and thus $C$ is closed. So $\mathbb{R}^\infty\setminus C$ is a neighborhood of $0$ which does not contain any $U_n$, showing that $(U_n)$ is not a neighborhood base at $0$.
(By the way, the fact that $\mathbb{R}^\infty$ is even a topological vector space at all is nontrivial. To prove the continuity of scalar multiplication and addition, you need to show that $\mathbb{R}\times\mathbb{R}^\infty$ and $\mathbb{R}^\infty\times\mathbb{R}^\infty$ also have the colimit topology with respect to their finite-dimensional subspaces. This follows from Theorem A.6 in Hatcher's Algebraic Topology, for instance.)