Dual Space of a Vector Space Question

Hello everyone I have a question about Dual Spaces. Let be an infinite dimensional vector space over , and let

be a subset of , which is linearly independent. Then we can define the unique linear functionals which give if 0 otherwise.

My question is can we choose the 's to be continuous? I was asked this question and I am not sure what the answer is. I take it if we have an infinite basis of X then we cannot have every continuous because then surely every linear functional would then be determined by those continuous functions, i.e. it is a linear combination of finitely many of them, hence is continuous. And there exist unbounded linear functionals (?). Any help with this would be appreciated! Thanks