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Let $V$ be a vector space over a field $F$ and $\{W_i\}_{i\in I}$ a collection of subspaces of $V$. Let $K$ be an extension field of $F$. If $I$ is finite then it is easy to see that $$\big(\bigcap_{i\in I} W_i\big)\otimes_F K=\big(\bigcap_{i\in I} W_i\otimes_F K \big).$$ Is this true also when $I$ is infinite?

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Yes, but I don't have as nice an argument as one might want, seemingly because "intersection" as categorical object has the natural maps going to it, rather than from, while the tensor product has maps going the opposite direction...?

But, nevertheless, the intersection has a basis (Axiom of Choice?), and extends to a basis of each $W_i$. It is not hard to prove that an "extension of scalars" of a free module produces a free module with "the same" generators, in the sense that they are images of the originals.

Thus, the intersection of the $W_i\otimes_F K$'s is no larger than the tensor product with the intersection.

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