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I am paraphrasing the definition of spanning given in Linear Algebra and Matrix Theory, 2nd Ed., by Ever D. Nering.

The set of all linear combinations of elements of any $\mathcal{A}\supset \mathcal{V}$ is called the set spanned by $\mathcal{A}$, and denoted by $\langle \mathcal{A}\rangle$. "It is part of this definition that $\mathcal{A}\subset \langle \mathcal{A}\rangle$."

The last part is a direct quote. It seems to me that $\mathcal{A}\subset \langle \mathcal{A}\rangle$ should be given as a result, rather than as part of the definition. Is there a reason not to consider it a logical consequence of the first part of the definition?

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    Well, formally it is a result, although a trivial one. The author is pretty much saying "It immediately follows from the definition that $\mathcal{A} \subseteq \left< \mathcal{A} \right>$". – 2017-02-26

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It looks like the author is confusing two common and equivalent definitions of $\langle\mathcal A\rangle$:

$\langle\mathcal A\rangle$ is the set of all linear combinations of elements of $\mathcal A$.

and

$\langle\mathcal A\rangle$ is the smallest subspace of $V$ that is a superset of $\mathcal A$.

In the second of these cases it is indeed part of the definition that $ \mathcal A \subseteq \langle\mathcal A\rangle$. In the first it is an easy consequence of the definition (every vector in $\mathcal A$ is a linear combination of itself with coefficient $1$), but not literally part of it.

(The second of the definitions is of course only a definition once we prove that there is a unique smallest such subspace -- but that isn't hard either).

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    I am vaguely familiar with the second usage from group theory. I will have to leave that for another day. I'm trying to shore up my linear algebra to help me understand Edwards's Advance Calculus of Several Variables which I am studying to shore up the skills necessary form mastering analytical dynamics and differential geometry which are needed for general relativity and quantum mechanics. Thanks for the guidance. It was helpful. – 2017-02-26
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Well, it follows from the definition, as the author probably means. If we have a set $A = \{a_1, a_2, \dots, a_n\}$, it is obvious that $\{1*a_1, \dots, 1*a_n\} = A \subset \langle A \rangle$, because we can write the elements of A as trivial linear combinations, as demonstrated.