Ambiguity of the Definition of Subspace

Question

In Hoffman's book, a subspace of a vector space is defined as follows:

Let $V$ be a vector space over the field $F$. A subspace of $V$ is a subset $W$ of $V$ which is itself a vector space over $F$ with the operation of vector addition and scalar multiplication on $V$.

I want to understand this definition more precisely, especially, trying to figure out whether the addition and scalar multiplication is on $V$ or $W$ when defining a subspace $W$ from an (existing) vector space $V$. With the notation of the addition $+_V: V \times V \rightarrow V$ and scalar multiplication $\cdot_{F, V}: F \times V \rightarrow V$, the definition is paraphrased as follows:

Given a vector space $(V, +_V, \cdot_{F, V})$ where $+_V$ and $\cdot_{F, V}$ are the addition and scalar multiplication on $V$, if a subset $W$ of $V$ is a vector space with $+_V, \cdot_{F, V}$, then we say $W$ is a subspace of $V$.

Question: Does the paraphrase precisely convey the idea of the original definition?

Some thoughts:

1. If the paraphrase is correct, can we also say $R^n$ with operations $+_{C^n}$ and $\cdot_{R, C^n} $ is a vector space? This looks a bit odd to me since we usually consider $+_{R^n}$ and $\cdot_{R, R^n}$ when defining a $R^n$ vector space. But looking up various resources regarding the operations within the definition of vector space, it seems there is no restriction for using an operation defined for a larger set (e.g., $C^n$) on a smaller (target) set (e.g.,$R^n$)

2. If the paraphrase is wrong, should we simply correct it as follows?

   Given a vector space $(V, +_V, \cdot_{F, V})$ where $+_V$ and $\cdot_{F, V}$ are the addition and scalar multiplication on $V$, if a subset $W$ of $V$ is a vector space with $+_W, \cdot_{F, W}$, then we say $W$ is a subspace of $V$. subspace of $V$."

   The only change from the previous paraphrase is "...vector space with $+_W, \cdot_{F, W}$...". But I don't think this is correct because $+_W, \cdot_{F, W}$ are not defined in the Hoffman's definition, and they are different from $+_V, \cdot_{F, V}$. Also, even they are defined, the paraphrase is not aligned with Hoffman's definition, in which the operations are said to be defined on $V$, i.e.,

   "...with the operation of vector addition and scalar multiplication on $\textbf{$V$}$."

Answer 1

The paraphrase is attempting to be more rigorous than the given definition, but it is not rigorous enough, so you are not really resolving your confusion.

Let's start with something a bit simpler: suppose you have sets $A$ and $B$, and a function $f\colon A\to B$. Given any subset $A'\subseteq A$, you can define the restriction of $f$ to $A'$, often denoted $f|_{A'}$. This is a function $f|_{A'}\colon A'\to B$ whose definition if $f|_{A'}(a') = f(a')$ for every $a'\in A'$. That is, the restriction of $f$ differs from $f$ by virtue of having a different domain, but othwerwise takes the same values.

If $g\colon A\to B$ is a function, $B'$ is a subset of $B$, and $g(A)\subseteq B'$, we can also define the co-restriction of $g$ to $B'$ (I am not aware of any standard or common notation for this, however), as $g'\colon A\to B'$, where $g'(a)=g(a)$ for every $a\in A$. When you care about both domain and codomain of a function, this is a function that is different from $g$ because it has different codomain.

If $\mathbf{V}$ is a vector space over $F$, then vector addition is a binary operation on $\mathbf{V}$, and hence a function $+_{\mathbf{V}}\colon\mathbf{V}\times\mathbf{V}\to\mathbf{V}$. We usually use infix notation for operations, rather than the traditional (in the United States, at any rate) prefix notation for functions. So we write "$\mathbf{x}+_{\mathbf{V}}\mathbf{y}$" to mean the result of applying the function $+_{\mathbf{V}}$ to $(\mathbf{x},\mathbf{y})$. That is, instead of writing the result of using the function as $+_{\mathbf{V}}(\mathbf{x},\mathbf{y})$, we write $\mathbf{x}+_{\mathbf{V}}\mathbf{y}$. Similarly, scalar multiplication is a function $\cdot_{F,\mathbf{V}}\colon F\times\mathbf{V}\to\mathbf{V}$, and we usually denote $\cdot_{F,\mathbf{V}}(\alpha,\mathbf{x})$ by $\alpha\cdot_{F,\mathbf{V}}\mathbf{x}$, or just $\alpha\mathbf{x}$.

If $\mathbf{W}$ is a subset of $\mathbf{V}$, then $\mathbf{W}\times\mathbf{W}$ is a subset of $\mathbf{V}\times\mathbf{V}$, so we can consider the restriction of $+_{\mathbf{V}}$ to $\mathbf{W}\times\mathbf{W}$. And since $F\times\mathbf{W}$ is a subset of $F\times\mathbf{V}$, we can consider the restriction of $\cdot_{F,\mathbf{V}}$ to $F\times\mathbf{W}$.

Suppose that we do that, and we have that whenever $\alpha\in F$, $\mathbf{x},\mathbf{y}\in \mathbf{W}$, we also have $\alpha\cdot_{F,\mathbf{V}}\mathbf{x}\in\mathbf{W}$ and $\mathbf{x}+_{\mathbf{V}}\mathbf{y}\in\mathbf{W}$. Then we can also consider the co-restrictions of $+_{\mathbf{V}}$ and $\cdot_{F,\mathbf{V}}$ to $\mathbf{W}$. Let us write this as $\overline{+_{\mathbf{V}}|_{\mathbf{W}\times\mathbf{W}}}$ (the co-restriction of the restriction to $\mathbf{W}\times\mathbf{W}$, and $\overline{\cdot_{F,\mathbf{V}}|_{F\times\mathbf{W}}}$.

With me so far? Okay, then the definition of "subspace" says:

If $(\mathbf{V}. +_{\mathbf{V}}, \cdot_{F\mathbf{V}})$ is a vector space over $F$, and $\mathbf{W}\subseteq \mathbf{V}$ is a subset of $\mathbf{V}$, then we say that $\mathbf{W}$ is a subspace of $\mathbf{V}$ if and only if $(\mathbf{W},\overline{+_{\mathbf{V}}|_{\mathbf{W}\times\mathbf{W}}}, \overline{\cdot_{F,\mathbf{V}}|_{F\times\mathbf{W}}})$ is a vector space over $F$.

(And the clutter probably lets you see why we usually try to paraphrase this rather than make the notation be very accurate...)

About your thoughts:

1. No. Note that while the restriction of $+|_{\mathbf{C}^n}$ to $\mathbb{R}^n$ does work (meaning, you can also do the co-restriction), the same is not true of $\cdot|_{\mathbb{C},\mathbb{C}^n}$: if you try to restrict it from $\mathbb{C}\times\mathbb{C}^n$ to $\mathbb{C}\times\mathbb{R}^n$, you can no longer also co-restrict it: for instance, $i\cdot (1,1,\ldots,1)$ does not lie in $\mathbb{R}^n$. So this restriction is not a "scalar multiplication" on $\mathbb{R}^n$. However, if you consider $\mathbb{C}^n$ as a vector space *over $\mathbb{R}$, then you do get that $\mathbb{R}^n$ is a subspace of $\mathbb{C}^n$ over $\mathbb{R}$.

2. The problem with your proposal in item 2 is that you do not explain what $+|_{\mathbf{W}}$ and $\cdot|_{F,\mathbf{W}}$ mean. You would need to say something like "where by $+|_{\mathbf{W}}$ we mean blah", etc.

The idea of the definition is that $\mathbf{W}$ should be a vector space in its own right, when you add vectors in $\mathbf{W}$ exactly the same way as you add them when you consider them as elements of $\mathbf{V}$. This is analogous to how when we add two integers as if they were fractions we get the same answer as when we add them as integers. We usually think of addition of rationals as extending the addition of integers (we already know how to add integers, we are not using that to explain how to add rationals). But if for some reason we knew how to add rationals but not how to add integers, we could ask whether when you add two fractions-correspond-to-integers we get a fraction-that-corresponds-to-an-integer, and if so restrict the addition-of-rationals to integers to get a way to add integers.

Answer 2

Question: Does the paraphrase precisely convey the idea of the original definition?

Yes.

The addition and scalar multiplication on the elements of the candidate subset are carried out in the original, ambient vector space. If all results fall again into the candidate subset, it is a subspace.

This is correctly mentioned in the Wikipedia as if under the operations of V, W is a vector space. The accepted answer above includes an interesting discussion rephrasing the same idea as domain restriction and co-restriction. But basically, the definition of subspace does not introduce any new operations.