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 $V$.

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    $\begingroup$ Could you perhaps provide some context? $\endgroup$ – user418131 Nov 28 '19 at 16:13
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    $\begingroup$ 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. $\endgroup$ – Alexander Geldhof Nov 28 '19 at 16:14
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    $\begingroup$ 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.) $\endgroup$ – runway44 Nov 28 '19 at 16:21
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    $\begingroup$ @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 $\endgroup$ – Masacroso Nov 28 '19 at 16:45
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    $\begingroup$ Could be just simply $\langle x,y \rangle_a = a\langle x,y \rangle$. It defines another inner product whenever $a>0$. $\endgroup$ – 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.

  • $\begingroup$ This makes a lot of sense, it's sometimes confusing when you're new to math whether some notation has special meaning or not. $\endgroup$ – Markus Amalthea Magnuson Nov 28 '19 at 17:26

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


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