My question may seem trivial but it's important that I know this. I know for a fact that a subspace of an inner product space is also an inner product space, but how about an arbitrary subset? Could I argue that since we are allowed to pick any subset, we could pick one that is not a subspace and since it is not a subspace it is not eligible as an inner product space?
Thank you!
On
An inner product space is by definition a vector space. Therefore, if a subset is not a subspace, then it cannot be an inner product space.
You can still talk about many properties that you might be interested in, such as perpendicularity and so forth.
For instance, you could take a subset $S$ of an inner product space $H$, and examine $\langle x,y \rangle$ for $x,y \in S$. For a subset of $S'$ you could also define $$(S')^\perp = \{ x \in S : \langle x,y \rangle =0 \text{ for all } y \in S\},$$ however $(S')^\perp$ would not necessarily be a subspace either.
A subset of an inner product space does not have to be an inner product space.
As you said, we can pick a subset that is not a subspace, hence not an inner product space.