Sum of Vector Spaces

Definition 1. Let $V$ be a vector space and $X$, $Y \subseteq V$ be subspaces. The sum of $X$ and $Y$ is

\[X + Y = \{x + y : x \in X, y \in Y\}.\]

Proposition 1. Let $V$ be a vector space. For any subspaces $X$, $Y \subseteq V$:

1. $X + Y$ is a subspace.

2. \[X + Y = \langle X \cup Y \rangle.\]

Proposition 2. Let $V$ be a vector space. For any finite-dimensional subspaces $X$, $Y \subseteq V$,

\[\dim(X + Y) = \dim X + \dim Y - \dim(X \cap Y).\]

Proposition 3. Let $V$ be a vector space. For any finite-dimensional subspaces $X$, $Y \subseteq V$,

\[\dim(X + Y) \leq \dim X + \dim Y.\]