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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.

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

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