# tensor products and intersections of subspaces

Given two real vector spaces $V$ and $U$ with subspaces $A, C \subset V$, $A\cap C \neq \{0\}$ and $B, D \subset U$, $B\cap D \neq \{0\}$, is it true that

$$(A\otimes B)\cap (C\otimes D) = (A\cap C)\otimes (B\cap D)$$

and if so, is there a proof available in a paper or textbook? It is stated to be true in a comment to a previous question Tensor Products and Intersections but no proof is given.

• Yes, have corrected Apr 29 '17 at 14:30
• I should add that the proof follows if the following is true: Consider the case where $A\cap C = \{0\}$, $B\cap D = \{0\}$, then $A\otimes B \cap C\otimes D = \{0\}$ Apr 29 '17 at 14:34

See e.g. Lemma 1.4.5 in the book

S. Dascalescu - C. Nastasescu - S. Raianu: "Hopf Algebras. An Introduction", Pure and Applied Mathematics, 2001

There the statement is proved under the assumption that $C\subseteq A$ and $D\subseteq B$: the general case easily follows from this.