Review of Finite-Dimensional Vector Spaces

# Review of Finite-Dimensional Vector Spaces

(1)
\begin{align} \quad x := \lambda_1 e_1 + \lambda_2 e_2 + ... + \lambda_n e_n \end{align}
• The Dual Basis for $E^*$ is the basis $\{ e_1^*, e_2^*, ..., e_n^* \}$ where for each $1 \leq i \leq n$ we define:
(2)
\begin{align} \quad \langle x, e_i^* \rangle := \lambda_i \end{align}

If $E$ is a finite-dimensional vector space then there is a unique topology for which $E$ because a Hausdorff and locally convex topological vector space.

If $E$ is a Hausdorff locally convex topological vector space and if $M$ is a finite-dimensional subspace of $E$ then $M$ is closed.