:= the intersection of subspace U such that U is a subspace of L containing M

Question

L is a linear space and set M is the set of points of L. The definition I put above is "the smallest subspace of L generated by M". The thing I don't understand in this definition is why do we need the brackets around M; how come we can't just say M the set of points which is the smallest subspace? Also U is not equal to the intersection of U?

Answer 1

The definition $$ \langle M \rangle := \bigcap \left\{U \,:\, U \text{ subspace of $L$, $M \subset U$} \right\} $$ doesn't define $M$, it defines the behaviour of the operator $\langle\cdot\rangle$. This operator maps every subset $M$ of $L$ to the smallest subspace of $L$ which contains $M$. That subspace is often called, as you state, the subspace generated by $M$.

It is certainly not true that $M = \langle M \rangle$. You always have that $\langle M \rangle \supset M$ (because $\langle M\rangle$ is the intersection of sets which all contain $M$), and $\langle M \rangle$ is always a subspace (because it's the intersection of subspaces). Thus, $M = \langle M \rangle$ holds exactly if $M$ is a subspace.