I'm studying vector spaces and I'm reading a proof where the authour uses the symbols
$$(\Rightarrow)$$
and
$$(\Leftarrow)$$
when proving a theorem. He doesn't use them in context, but rather before starting a part of the proof.
How should I read this? Does it have anything to do with symbols and notation in logic?
Seems Gerry's answer is the case. I'll just add the example:
Proposition 1.10
Let $V$ be a $K$-vector space, and let $S \subseteq V$. Then $S$ is a subspace of $V$ if and only if:
- $\mathbf{0} \in S$
- $\mathbf{v},\mathbf{w} \in S\Rightarrow \mathbf{v}+\mathbf{w}\in S$
- $\lambda \in K,\mathbf{V}\in S\rightarrow \lambda \cdot \mathbf{v}\in S$
Proof
$(\Rightarrow)$ It is immediate [sigh] to verify that if $S$ is a subspace of $V$ then $1$,$2$ and $3$ hold.
$(\Leftarrow)$
$1.$ ensures $S \neq \emptyset $
$2.$ implies that $+$ is an operation in $S$.
$3.$ implies that $\cdot$ is an action.
Asociativity and conmutativity are deduced from their validity in $V$, the neutral element from $1.$, and the inverse additive from $3$. The properties of the action are deduced from their validity in $V$.
Best Answer
He's probably proving a statement of the form "$A$ if and only if $B$", and he's using the first arrow to indicate he's starting the proof of the $A$ implies $B$ part, and the second arrow to indicate he's starting the proof of the $B$ implies $A$ part.