Review of Finite-Dimensional Vector Spaces

# 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:

\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:

\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.**