# Definition (as a Vector Space)
A Euclidean Space of dimension $n$ is the Vector Space on $\mathbb{R}^{n}$ over Field $\mathbb{R}$ where
- for $x, y \in \mathbb{R}^{n}$, $x + y := (x_{1} + y_{1}, …, x_{n} + y_{n})$
- for $x \in \mathbb{R}^{n}$ and $c \in \mathbb{R}$, $cx := (cx_{1}, …, cx_{n})$
# The Euclidean Space is a Vector Space
Proof: TODO. This is not very difficult, just requires running through the definition of a Vector Space.
# Definition (as a Inner Product Space)
If we endow the Euclidean Space (as a Vector Space) with the Dot Product, it becomes an Inner Product Space. This gives rise to a Normed Vector Space and Metric Space because Inner Products induce Norms and Norms induce Metrics.
# Definition (as a Topological Space)
Taking the Euclidean Space as the Metric Space described above, we can endow it with the Metric Topology. Although it might seem that this Topological Space depends on the Inner Product used above, All Norms are Equivalent on Finite Dimensional Spaces, so any Norm works.