# Necessity for the equality of two subspaces

Let $R$, $S$, and $T$ be subspaces of a vector space.
I was able to show that $(R \cap T)+(S \cap T) \subseteq (R+S) \cap T$.

So my question now is: When will equality happen without using the other inclusion?

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What does "without using the other inclusion" exactly mean? –  manado Dec 11 '12 at 13:30
Since it was established that $(R \cap T)+(S \cap T) \subseteq (R+S) \cap T$. I mean not to use this inclusion $(R \cap T)+(S \cap T) \supseteq (R+S) \cap T$ to establish the equality. In other words, what should be the sufficient hypothesis for it to be equal. Thanks. –  Philip Benj Marcoby Eragon Dec 11 '12 at 13:48
If your vector space is a finite dimensional vector space with an inner product, the two sides are the same. –  user1551 Dec 12 '12 at 0:40