What does the notation ${\langle x, y \rangle}_a$ with $a \in \mathbb{R}$ mean?

Question

I know that ${\langle x, y \rangle}$ means the inner product but I've stumbled upon the notation ${\langle x, y \rangle}_a$ with $a \in \mathbb{R}$ and I can't figure out what it means. Usually what's in the subscript isn't a number, but the denotion of some vector space, e.g. $V$.

The context is this problem from an exam in the introductory course in linear algebra at our university:

Let ${\langle,\rangle}_1$ and ${\langle,\rangle}_2$ be two inner product structures on a finite-dimensional vector space $V$. Show that there is a linear map $T: V \to V$ such that

${\langle x,y \rangle}_1$ = ${\langle T(x), y \rangle}_2$

for all $x$ and $y$ in $V$.

Answer 1

In your context the notation $\langle f,g \rangle_1$ or $\langle f,g \rangle_2$ is just a way to name two distinct inner products, this is all. The numbers doesn't have a "mathematical" meaning, it just a name, a tag.

We could say also that there are two inner products, represented as $\langle f,g\rangle_{\text{ foo }}$ and $\langle f,g \rangle_{\text{ bar }}$ to distinguish them.

Answer 2

The answer @Masacroso gave is probably right. I should mention subscripts on either side may instead be labels for the vectors themselves. However, this is usually used only in bra–ket notation, where we replace the comma with a pipe. In other words, $\langle x|y\rangle_a$ could be $\langle x|z\rangle$ with $|y\rangle_a:=|z\rangle$.