Discontinuous functionals on infinite dimensional topological vector space [closed]

It is easy to see that every infinite dimensional normed vector space has discontinuous functionals. My question is: Is this also true for general topological vector spaces?

Every linear functional on the vector space $X$ is continuous relative to the topology $\sigma(X,X^*)$, where $X^*$ is the algebraic dual of $X$.

No, it is not true for general TVS. Given any vector space $X$, it is possible to construct a Hausdorff locally convex topology $T$ on $X$ that has the property that any convergent sequence in $(X,T)$ is contained in some finite-dimensional subspace of $X$. This guarantees that every linear functional on $(X,T)$ is continuous. A base of neighbourhoods of zero for such a topology can be taken as the family of all absolutely convex absorbing subsets of $X$. (We can choose $X$ to be initially infinite-dimensional)

Such a topology is not metrizable. As already mentioned in the comments, every infinite-dimensional metrizable TVS admits a discontinuous linear functional.