Meaning of notation $v \mapsto ( x \mapsto f(x,v) )$

Question

I came across this notation in this wiki article.

Can anyone tell me the meaning of this notation? What exactly is happening here?

$v \mapsto ( x \mapsto f(x,v) )$

What I understand here is $x, v \in V$ and $V^*$ is its dual.

Am I correct, if I say, first vector v is fixed and

then $\forall x, x \mapsto f(x,v)$?

So that $f( . , v) \in V^*$, where dot will be filled by $x$ ?

Thanks in advance.

Answer 1

The notation $v \mapsto ( x \mapsto f(x,v) )$ means that a vector $v \in V$ is sent to the element $f(\cdot , v) \in V^*$.

Answer 2

In general the \mapsto arrow $\mapsto$ is a way of defining a function without naming it. It's similar to the lambda construction in Scheme and Python, and I believe Maple uses ->. So for example, you might define $$f(x) = x^2$$ or you could write the same definition as $$f = x \mapsto x^2$$ So the notation $$F = v \mapsto (x \mapsto f(x,v))$$ from that Wikipedia page could also be written as $$F(v) = x \mapsto f(x,v)$$ or using the $\cdot$ notation, $$F(v) = f(\cdot, v)$$ Since $f$ in the Wikipedia article is supposed to be a bilinear form, if you fix one of its parameters (fill in $v$) you get a linear function from vectors to scalars.

See Regarding the notation $f: a \mapsto b$