Let $V$ be a vector space and $X$, $Y \subseteq V$ be subspaces. The sum of $X$ and $Y$ is
\[X + Y = \{x + y : x \in X, y \in Y\}.\]
Let $V$ be a vector space. For any subspaces $X$, $Y \subseteq V$:
- $X + Y$ is a subspace.
- $X + Y = \langle X \cup Y \rangle$.
Let $V$ be a vector space. For any finite-dimensional subspaces $X$, $Y \subseteq V$,
\[\dim(X + Y) = \dim X + \dim Y - \dim(X \cap Y).\]
As a corollary:
Let $V$ be a vector space. For any finite-dimensional subspaces $X$, $Y \subseteq V$,
\[\dim(X + Y) \leq \dim X + \dim Y.\]