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I have a few questions regarding Axler's proofs that concern "smallest containing subspaces": in particular, 1.39.

I understand we want to show that $U_1 + \cdots + U_m$ = the smallest subspace of V containing $U_1, \cdots, U_m$ by showing that each set is a subset of the other set. What I don't understand, however, is why Axler says "Conversely, every subspace of V containing $U_1 \cdots U_m$ contains $U_1 + \cdots + U_m$)" because isn't the point to compare $U_1 + \cdots + U_m$ with the SMALLEST subspace, not an arbitrary subspace?

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José Carlos Santos
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    To show it is the smallest, you want to show that (i) it is a subspace containing all of $U_1,\ldots,U_n$; and (ii) it is contained in every subspace that contains all of $U_1,\ldots,U_n$.

    For general comments about this sort of thing, see this answer.

     – Arturo Magidin Nov 30 '20 at 22:03

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If $V$ is a subspace containing $U_1,\ldots,U_m$, then $V\supset U_1+U_2+\cdots+U_m$. In particular, the smallest subspace containing $U_1,\ldots,U_m$ will also contain $U_1+U_2+\cdots+U_m$. But $U_1+U_2+\cdots+U_m$ is a subspace itself, which contains $U_1,\ldots,U_m$. So, $U_1+U_2+\cdots+U_m$ contains the smallest subspace containing $U_1,\ldots,U_m$, and therefore they're equal.

José Carlos Santos
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