# Quotient Space Definition: Modulo Non-Subset

Suppose $$Z$$ is a linear space and $$Y \subseteq Z$$ is a linear subspace. Then we can define the quotient space $$Z\mid_Y$$, i.e. $$Z$$ modulo $$Y$$, as follows:

$$Z\mid_Y = \{ [z]_Y \mid z \in Z \}$$

Where $$[z]_Y$$ is the equivalence class of $$z$$ for the relation $$\sim^Y$$ defined on $$Z$$ by:

$$x \sim^Y y \equiv x - y \in Y$$

This agrees with the Wikipedia definition of quotient space.

Question: Suppose $$X_0 \subseteq Z$$ is another linear subspace of $$Z$$. Then how do we define the quotient space $$X_0\mid_Y$$? We can't just apply the previous definition because, in general, we do not have that $$Y \subseteq X_0$$. However, I believe we can just relax that restriction and do this:

$$X_0\mid_Y = \{ [x]_Y \mid x \in X_0 \}$$

I'm 99% percent sure that this definition is the correct one, but hoping someone can point me to a definition of quotient space that doesn't require $$Y$$ to be a subset of the space being "divided"?

What's slightly odd to me is that the equivalence classes in this resulting quotient space are not necessarily subsets of $$X_0$$, so it goes against the intuition that $$Y$$ is somehow "dividing" up $$X_0$$.

### Alternative Formulation: Cosets

As per this question, we can also define the quotient space $$Z\mid_Y$$ as follows:

$$Z\mid_Y = \{ z + Y \mid z \in Z \}$$

Where $$z + Y = \{z + y \mid y \in Y\}$$ is the coset of $$Y$$ containing $$Z$$. In this case, what I'm asking is can we define the quotient space for $$X_0$$ in general, without necessarily having $$Y \subseteq X_0$$, as follows:

$$X_0\mid_Y = \{ x + Y | x \in X_0 \}$$

This definition seems natural to me - the only unnatural part is the term "quotient" in the term "quotient space", and the division-like notation in $$X_0\mid_Y$$, because we're not necessarily dividing up $$X_0$$, rather we're sort of mapping it over to a set of cosets. But I suppose since the term "quotient" makes sense for the case when $$Y \subseteq Z$$, maybe we just overload it to apply also to this case? Or is this called something else?

### Background

I came across use of this version of quotient at the end of this video, where the professor equates the completion of a space to the closure of the "quotient" of its embedded space:

VIDEO: Completing a normed linear space. Time: 1:01:33.

### EDIT: Reason for my Confusion

I initially thought that:

$$X_0\mid_Y = X_0\mid_{Y \cap X_0}$$

The reasoning being:

1. $$Y$$ and $$X_0$$ are linear spaces
2. The intersection of linear spaces always yields a linear space
3. $$Y \cap X_0 \subseteq X_0$$, so we can apply the original definition

However, in general, this is not the case. That is, there exists some example such that:

$$X_0\mid_Y \neq X_0\mid_{Y \cap X_0}$$

In fact, the example in the linked lecture video is such an example:

$$X_0$$ is the set of constant sequences of the form $$(x, x, x, \dots)$$. $$Y$$ is the set of sequences $$(y_1, y_2, y_3, \dots)$$ which are equivalent to $$(0, 0, 0, \dots)$$ in the sense that limit of the norms of elements $$y_i$$ converge to zero. So in that case $$Y \cap X_0$$ is the singleton set:

$$Y \cap X_0 = \{(0, 0, 0, \dots)\}$$

Which certainly is not equal to $$Y$$, as long as there is some non-zero element $$x \in X$$, because we can define $$s = (x, 0, 0, \dots)$$ and we know that $$s \in Y$$ but clearly $$s \notin Y \cap X_0$$ because $$s \neq (0, 0, 0, \dots)$$.

So then, for example, to show that

$$X_0\mid_Y \neq X_0\mid_{Y \cap X_0}$$

We just have to show that the equivalence classes differ for one element. Choose $$0$$ to be that element:

$$_Y = Y \neq Y \cap X_0 = _{Y \cap X_0}$$

So the quotient spaces are not equal.

Once you chose your linear space $$Z$$ and your subspace $$Y\subset Z$$, you have a linear (and onto) map $$\pi:Z\to Z|_Y$$ given by $$z\mapsto [z]_Y$$. Since a linear map sends linear subspaces to linear subspaces, you can see that both of your definitions agree with $$X_0|Y=\pi(X_0)$$. An equivalent way to define this would be $$X_0|_Y:=X_0|_{Y\cap X_0}$$, since $$X_0\cap Y$$ is then a linear subspace of $$X_0$$.
• The quotient spaces are not equal in the sense that they have the same elements, but rather in the sense that they are isomorphic: the linear map $L:X_0|_{Y\cap X_0}\to \pi(X_0)$ given by $[x]_{Y\cap X_0}\mapsto[x]_Y$ is onto by definition, and injective since $L([x]_{Y\cap X_0})=0$ implies that $[x]_Y=0$, then $x\in Y$, so $x\in X_0\cap Y$ and thus $[x]_{Y\cap X_0}=0$. Dec 7, 2021 at 17:09
• Thanks. The isomorphism makes sense. I guess though the heart of the question is this. Strictly speaking, do we define: $X_0|_Y:=X_0|_{Y\cap X_0}$. Or do we define it as the image under $L$: $X_0|_Y:=L(X_0|_{Y\cap X_0}) = \pi(X_0)$. I'm assuming the latter, based on the lecturer's comments. Otherwise his proof doesn't make sense. But I can't find an official definition anywhere. I guess the question is more focused on is there a reference where this object is officially defined - the quotient of a space modulo some non-subspace, both in the context of a larger space that contains both. Dec 7, 2021 at 18:08
• Yes, I agree! I want to add that since the isomorphism is canonical, it can be made precise to think about these spaces as the same spaces in some sense (in a categorical way). Also think about how we identify $\mathbb{R}$ and $\mathbb{R}^1$ where the latter is rigorously defined as the set of functions $f:\{1\}\to\mathbb{R}$ endowed with natural operations, I think these are problems of same nature. Dec 7, 2021 at 18:17
• Cool. I think I see your answer in a slightly different light now. I think what you're saying is "While the thing defined as $X_0 \mid_Y$ is not officially a quotient space, as per the Wiki definition, $X_0 \mid_{Y \cap X_0}$ is, and moreover, it's isomorphic to the first, so they're practically the same thing for all intents and purposes." Dec 8, 2021 at 8:24