Let X be an infinite dimensional Banach space. Let $\{e_1, e_2, ...\}$ be a linearly independent set of vectors whose linear span is dense in X. Define $\lambda_i \in X^{*}$ by $\lambda_i (e_j) = \delta_{i, j}$. Prove or disprove that the linear span of $\{\lambda_1, \lambda_2, ...\}$ is dense in $X^{*}$.
I want to say this is false. Am I correct in saying that for infinite-dimensional Banach spaces, the cardinality of $X^{*}$ is always larger than the cardinality of X? If so, can I use this here?
No. Just consider any Hilbert space as a counterexample, where Riesz representation theorem guarantees that $X$ and $X^*$ have the same cardinality.
The proposition the question asks about, is indeed false. Counterexample: $X = \ell^1, X^*=\ell^\infty$, $$e_i = (0, \cdots, 0, \underbrace{1}_{i\text{-th position} }, 0, \cdots).$$ As can be verified directly, the span of $\{e_i\}$ is dense in $\ell^1$. However, the dual functionals $$\lambda_i = (0, \cdots, 0, \underbrace{1}_{i\text{-th position} }, 0, \cdots)$$ is not dense in $\ell^\infty$. For example, $(1, \cdots, 1) \in \ell^\infty$ can not be approximated by $\text{Span} \{\lambda_i\}$; the $\ell^\infty$-distance is always 1.