Linear Algebra Done Right 2nd Edition – Chapter 1 Exercise 8

Chapter 1 Exercise 8.
Prove the intersection of any collection of subspaces of V is a subspace of V .

Proof:
Let $X_1, X_2,...$ be subspaces of $V$.
Then $O_V \in X_q \forall q$. Thus,

(1) $O_V \in X_1 \cap X_2 \cap ...$

Therefore, $X_1 \cap X_2 \cap ... \neq \emptyset$.
Suppose $X_1 \cap X_2 \cap ... = \{ x_1, x_2, ... \}$. Then $x_i, x_j \in X_q \forall i, j, q$.
Since $X_q$ is a subspace, $x_i + x_j \in X_q \forall q$. Thus

(2) $x_i + x_j \in X_1 \cap X_2 \cap ...$

Since $X_q$ is a subspace, $ax_i \in X_q \forall q, i$ and scalar $a$. Thus

(3) $ax_i \in X_1 \cap X_2 \cap ...$

By 1, 2 and 3, $X_1 \cap X_2 \cap ...$ is a subspace of $V$.

$\square$