Intersection of infinite subspaces

Question

Let $V$ be an infinite-dimensional vector space over a field $F$ and let $W_1$ and $W_2$ be infinite-dimensional subspaces. Prove or disprove: if $W_1\cap W_2\neq\{0\}$, then $W_1\cap W_2$ has infinite dimension.

I have no idea where to start with this question. I have a hard time working with infinite-dimensional spaces since they rarely come up in examples in this class.

I do know that $\dim W_1+\dim W_2=\dim(W_1\cap W_2)+\dim(W_1+W_2)$, but since $\dim(W_1+W_2)=\dim W_1=\dim W_2=\infty$, this is not very helpful.

Answer 1

False.

Let $V = \mathbb R^{\mathbb N}$ (the vector space of all real sequences).

Let $W_1 = \{(a_1, a_2, 0, a_4, 0, a_6, \ldots): a_i \in \mathbb R\}$ (sequences with odd terms $0$ with the exception of the first term).

Let $W_2 = \{(a_1, 0, a_3, 0, a_5, 0, \ldots): a_i \in \mathbb R\}$ (sequences with even terms $0$).

Answer 2

Let $X$ and $Y$ be two infinite dimensional vector spaces. Inside of $V = X \oplus Y \oplus F$ we can consider $W_1 = X \oplus 0 \oplus F$ and $W_2 = 0 \oplus Y \oplus F$.

Their intersection is $0 \oplus 0 \oplus F$, which is finite dimensional and non zero.

Answer 3

It is false. Let $V$ be the vector space of all functions from $\mathbb Z$ into $F$, let$$W_1=\left\{f\in V\,\middle|\,f(n)=0\text{ if }n\geqslant0\right\}\text{ and }W_2=\left\{f\in V\,\middle|\,f(n)=0\text{ if }n\leqslant0\right\}.$$Then $\dim(W_1\cap W_2)=1$.