#math/linear-algebra #flashcards/math A `linear functional` on $V$ is %%::%% a linear map from $V$ to $\mathbf{F}.$ In other words, a linear functional is an element of $\mathcal{L}(V,\mathbf{F})$. A `dual space` of $V$, denoted $V'$, is %%::%% the vector space of all linear functionals on $V$. In other words, $V'=\mathcal{L}(V,\mathbf{F})$. # Dimension of $V'$ Suppose $V$ is finite-dimensional. Then $V'$ is also finite-dimensional and $ \text{dim }V'=\text{dim }V. $ **Proof:** $\text{dim }V'=\text{dim }\mathcal{L}(V,\mathbf{F})=(\text{dim V})(\text{dim }\mathbf{F})=\text{dim }V$.