Review of Finite-Dimensional Vector Spaces

• Recall from The Dual Base for E* of a Finite-Dimensional Vector Space E page that if $E$ is a finite-dimensional vector space with $\mathrm{dim}(E) = n$ and if $\{ e_1, e_2, ..., e_n \}$ is a basis for $E$, then every $x \in E$ can be written uniquely in the form:

(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}

• On the Finite-Dimensional Vector Spaces Have a Unique Locally Convex and Hausdorff Topology page we proved the following important result:

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.

• Lastly, on the Finite-Dimensional Subspaces are Closed in a Hausdorff LCTVS page we proved that:

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