Vector space

Define

… as the set

“\(V\) defined over the field \(F\)”

that contains the zero vector, so

\[\begin{equation*}
  \vec{0} \in V,
\end{equation*}
\]

in conjunction with the vector addition operation

\[\begin{gather*}
  \forall \vec{u}, \vec{v} \in V, \\[1ex]
  \vec{u} + \vec{v} \in V
\end{gather*}
\]

that satisfies all properties listed here

\[\begin{gather*}
  \forall a \in F, \> \forall \vec{v} \in V, \\[1ex]
  a \vec{v} \in V
\end{gather*}
\]

that satisfies all properties listed here.

… by \(V^n\) to communicate that \(\forall \vec{v} \in V^n\), \(\vec{v}\) has exactly \(n\) components.