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

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 $$V$$.

• Could you perhaps provide some context? – user418131 Nov 28 '19 at 16:13
• Possibly $\langle x, y \rangle_a := [\sum (x_i \cdot y_i)^{a} ]^{a^{-1}}$ (in particular check what this gives for $a = 2$), but it is hard to know without context. – Alexander Geldhof Nov 28 '19 at 16:14
• OP, you don't provide any context, which suggests to me you were confident the notation is standard enough your readers wouldn't need any, but I don't think it is and you'll have to provide. @AlexanderGeldhof What's special about $a=2$? (It doesn't correspond to the $2$-norm's corresponding inner product.) – runway44 Nov 28 '19 at 16:21
• @Alexander as you defined $\langle x,y \rangle_a$ it cannot be an inner product for $a\neq 1$ because it would not be linear in it first argument – Masacroso Nov 28 '19 at 16:45
• Could be just simply $\langle x,y \rangle_a = a\langle x,y \rangle$. It defines another inner product whenever $a>0$. – azif00 Nov 28 '19 at 17:02

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.
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 $$|y\rangle_a:=|z\rangle$$.