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  1. Let $(G,\circ)$ be a group, $(H,\circ|_{H})$ is its subgroup.
  2. Let $(X,d)$ be a metric space, $(Y,d|_{Y\times Y})$ is its metric subspace.

In both case, the right notation seems to be what I wrote(i.e. use a restriction to the original function). However, I'm not sure if someone just write $(H,\circ)$ is a subgroup of $(G,\circ)$ and $(Y,d)$ is a metric subspace of $(X,d)$. Well, this is not consistent of the definition of subgroup or subspace, because the domain of the function(e.g. $\circ$ and $d$), are all exceed the range of these sub-structure(the subset $H$) or sub-space(the subset $Y$) themselves. So which notation is in common used?

Would the restriction style too lengthy? Or would the non-restriction style too informal?

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    It's always the proper restriction, it's only slight abuse of notation to not indicate that. – 2017-02-20
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    Do yourself a favour and get used to this abusive notation. There will come a point in mathematics (algebraic topology and differential geometry come to mind) where you need to be comfortable with this abuses of notation as otherwise the notation becomes too heavy. – 2017-02-20
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    @noctusraid So to confirm what you said, it is the case that it is pretty common that people write $(Y,d)$ or $(H,\circ)$ than $(Y,d|_{Y\times Y})$. And the convenience of doing so will become clearer in more advanced topics right? If most people do so, I'm glad to use such symbol. :) – 2017-02-20
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    I'd say so, yes! :) – 2017-02-20

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