Notation question: What does $\langle X, - \rangle$ exactly mean?

Question

Let $ \langle \cdot, \cdot \rangle$ be an inner product on $\mathbb{R}^n$

Then according to my course notes

$$X \mapsto X^{b} = \langle X, - \rangle$$

is an isomorphism from vector fields to one-forms.

But my simple quesion is, what does the '$-$'-symbol represent?

Also what does the flat $b$ stand for as in $X^{b}$ ?

Answer 1

The $-$ is just a place holder indicating that we can input arbitrary vector fields into the second location in the inner product with $X$ fixed in the first location. To be concrete, $X^\flat$ denotes the $1$-form such that when evaluated on a vector field $Y$ produces the function $\langle X,Y \rangle$: $$X^\flat(Y) = \langle X,Y \rangle.$$