# What does the dash mean in “$v \mapsto \langle v, -\rangle$”?

This quote is taken from a math.SE answer (https://math.stackexchange.com/a/622695/816960):

By the way, for Euclidean vector spaces, there is a canonical isomorphism, given by $$V \to V^*$$, $$v \mapsto \langle v, -\rangle$$.

What does the dash in this inner product mean? I've never seen this notation before? I assume it's some kind of placeholder, but for what I can't guess.

• How do you write an element of $V^*$? – Mark Bennet Mar 6 at 17:47
• I would say it's the linear form associated to \$vˆin an inner product space. – Bernard Mar 6 at 17:58

An element of $$V^*$$ is a linear map on $$V$$, ultimately a function $$V \to F$$. The notation $$v \mapsto \langle v, -\rangle$$ is a shorthand for $$v \mapsto (w \mapsto \langle v, w \rangle)$$, producing a member $$w \mapsto \langle v, w \rangle$$ of $$V^*$$ given a member $$v$$ of $$V$$.