Justifications for the Definition of the Dimension of the Quotient Space

Question

Let $X$ be a linear space and $Y$ be a linear subspace of X. The set $X/Y$ consisting of equivalence classes (mod $Y$) is called the quotient space of $X$ mod $Y$.

It is defined that: $$codim (Y)=dim(X/Y)$$

My questions are:$$$$ 1. Does this definition still have a meaning if both $Y$ and $Y/X$ are of infinite dimensions?$$$$ 2. According to the definition, $$dim(X/Y)+dim(Y)=dim(X);$$ Why is this true (in terms of both finite and infinite dimensional spaces)?

Answer 1

Well! Ambiguity may always happen for example take:$$X=\{(x_1,...,x_n)|\forall n\in\Bbb N\qquad,\qquad x_1,...,x_n\in\Bbb R^n\}$$and for some $k\in\Bbb N$ take$$Y_k=X-\{(x_1,...,x_k)\in X\}$$therefore $dim(X)$ and $dim(Y)$ are both $\infty$ and $dim(X/Y)=\infty-\infty$ which is an ambiguity however it is still true.

Answer 2

If $B_X=\{x_i+Y\,:\,i\in I\}$ is a basis of $X/Y$ for some $x_i\in X$ and $B_Y$ is a basis of $Y$, one can see that $B:=\{x_i\,:\,i\in I\}\dot\cup B_Y$ is a basis of $X$ (in finite and infinite dimensional cases). In the finite dimensional case it follows that

$\dim X=|B_X|+|B_Y|=\dim X/Y+\dim Y$

where the dimensions are natural numbers.

To work in infinite dimensions, we use cardinal numbers. On the family of all sets we can define two sets to be equivalent iff there is a bijection between them. This is an equivalence relation, and the equivalence classes are the cardinal numbers. For a set $P$ we denote its cardinal number (cardinality) as $|P|$. If $P$ has $n\in\mathbb N$ elements, its equivalence class consists of all sets of $n$ elements, therefore we often write $|P|=n$ to refer to that equivalence class, which coincides with the old definition of $|P|$ being the number of elements.

Addition of cardinal numbers is defined using the disjoint union, that means $|P|+|Q|:=|P\dot\cup Q|$ when the sets are disjoint, otherwise pick some $Q'$ with same cardinality as $Q$ that is disjoint to $P$ and define $|P|+|Q|:=|P\dot\cup Q'|$.

The dimension of a vector space then is defined as the cardinality of any basis (which is the same for all bases).

From the disjoint union equation above it follows that

$\dim X=|B|=|\{x_i\,:\,i\in I\}\dot\cup B_Y|=|\{x_i\,:\,i\in I\}|+|B_Y|=|B_X|+|B_Y|=\dim X/Y+\dim Y$

Therefore all your definitions make sense in all dimensions.