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A function $T:V \rightarrow W$ is additive if $T(x+y) = T(x) + T(y)$ for every $x, y \in V$. Prove that there exists an additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.

My attempt: Let $T$ be the function $T: \mathbb{R}$ (over the field $\mathbb{Q}$) $\rightarrow \mathbb{R}$ (over the field $\mathbb{R}$). The set $\{1, \sqrt{2}\} \subseteq \mathbb{R}$ is linearly independent for the vector space $\mathbb{R}$ over $\mathbb{Q}$. Then, there must exist a linearly independent set $W \subseteq \mathbb{R}$ (over $\mathbb{Q}$) such that $\{1, \sqrt{2}\} \subseteq W \subseteq \text{span}(W) = \mathbb{R}$ (over $\mathbb{Q}$).

I have been told that the function defined as $T(1) = 1$ and $T(w) = 0$ for all $w \in \text{span}(W) \setminus \{1\}$ is additive but not linear, but I cannot see why this is? I can see why it is not linear, clearly $T(\sqrt{2} \cdot 1) = 0$ but $\sqrt{2} T(1) = \sqrt{2}$. But, why is $T$ additive? For example, $T(1+1) = T(2) =0$ but $T(1) + T(1) = 1+1 = 2$? Is there a mistake somewhere?

elbarto
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    Be careful, you want the $\mathbb Q-$homogeneity ! Therefore $T(\sqrt 2 \cdot 1)\neq \sqrt 2 T(1)$ since it's not $\mathbb R-$homogeneous. – Surb Jul 31 '17 at 12:46

3 Answers3

17

You can indeed construct an additive, non-linear function $T:\Bbb R\to \Bbb R$ (as vetor spaces over $\Bbb R$, since additive maps of $\Bbb Q$-vector spaces necessarily become linear automatically) by starting with what you have: $T(p) = p$ for $p\in \Bbb Q$, and $T(r) = 0$ for all $r$ with $r/\sqrt2 \in \Bbb Q$. However, constructing it is not easy, and requires the so-called axiom of choice in order to be finalised.

So, we start out with what we already have: some values that $T$ sends to themself, and some values that $T$ sends to $0$. By additivity, this forces $T$ to be defined for a slightly greater class of numbers, namely $T(p + q\sqrt2) = p$, for all rational $p, q$.

But we haven't defined $T$ for all real numbers yet. Note that whatever else we define $T$ to be, it already cannot be linear. So we just build on it as best we can and ensure that it stays additive. We do this by picking a number for which $T$ isn't already defined, like $\pi$ or $\sqrt 3$ or $\ln 2$ or $e$ or anything else. I'll pick $\pi$ for now. We can make $T(\pi)$ to be whatever we want it to be. I'll pick $3$ because I like $3$. By additivity, $T$ is now forcibly defined to be $T(p+q\sqrt2+r\pi) = p+3r$ for any rational $r$.

However, we are not done definining $T$ yet; there are still many more real numbers to cover. And this is where the axiom of choice comes in: there is no finite way, or even countably infinite way to finalise this definition of $T$. We just have to keep going and going, picking one new real number each step, until we've covered the entire real number line, which requires an uncountably infinite number of steps. The axiom of choice is exactly what allow us to say that even though we cannot finalise our $T$, some finalised $T$ does indeed exist somewhere out there.

Arthur
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8

You need some key elements here.

First of all, note that if we consider $\mathbb R$ as a vector space over $\mathbb Q$, then it is infinite dimensional. This means that a "Hamel" basis $E$ for $\mathbb R$ over $\mathbb Q$ is at least countably infinite, and exists by the axiom of choice. Note : Without the axiom of choice, this is a rather interesting question, one that I don't know much about.

Choose two elements $e_1,e_2$ from this Hamel basis. Define the "linear transformation" $f$ from $\mathbb R \to \mathbb R$ as vector spaces over $\mathbb Q$, by defining $f(e_1) = 1$ and $f(e) = 0$ for all $e \neq e_1 \in E$. Note that because we are extending $f$ from it's basis construction, it obviously is additive, but then it isn't of the said form, since $f(e_2) = 0 \neq \frac{e_2}{e_1}f(e_1)$.

Hence $f$ is additive, but not linear.

(Read discussion below for field-related query)

Sarvesh Ravichandran Iyer
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  • Yes, that is right. It is to be corrected. Indeed, I meant the base fields to be the same, but only indexed the first one to indicate that the base field of the second will be the same. – Sarvesh Ravichandran Iyer Jul 31 '17 at 13:13
  • Some clarification: (correct me if I've misunderstood your intent) $f$ is linear when $\Bbb R$ is seen as a vector space over $\Bbb Q$ (any additive function over $\Bbb Q$-vector spaces is automatically linear), but not when seen as a vector space over $\Bbb R$. However, additivity is independent of the base field of the vector space, and thus $f$ is still additive. – Arthur Jul 31 '17 at 13:16
  • Aslo, do you mean to define $f(e_1) = 1$? Because the way it stands now, your $f$ is actually the constant zero function. – Arthur Jul 31 '17 at 13:19
  • Oh, many errors. Let me edit it and get back to you. Thank you. – Sarvesh Ravichandran Iyer Jul 31 '17 at 13:19
  • Okay, so after corrections : You must clarify what this statement means, because it's confusing me : "any additive function over $\mathbb Q$-vector spaces is automatically linear". What is the difference between linear and additive, in the context of $\mathbb Q$- vector spaces? Because this notion of $f(x) = kx$ seems to make sense only in $\mathbb R$, right? Again, I must be wrong, so please correct me, it can only be helpful. – Sarvesh Ravichandran Iyer Jul 31 '17 at 13:25
  • Say we have an additive function $g:X\to Y$, where $X$ and $Y$ are vector spaces over $\Bbb Q$. Then we necessarily have $f(nx) = nf(x)$ for natural numbers $n$. We can then expand to the same relationship for integers $n$ by noting that $f(x + (-x)) = f(0) = 0$. Then we note that $f(x) = f(\frac{nx}{n}) = nf(\frac xn)$, so $f(\frac xn) = \frac1nf(x)$. Thus $f(nx) = nx$ is valid for all rational $n$. But together with $f(x+y) = f(x) + f(y)$, this implies linearity. The reason that additive functions aren't necessarily additive for real vector spaces is that you can't reach $\pi$ the same way. – Arthur Jul 31 '17 at 13:33
  • Okay, so additivity does not depend upon the base space, since it only involves the operation $+$, and elements from the space. However, the problem with linearity is that it brings scalars from the base field into the question, and these are the real problem when the base field changes. – Sarvesh Ravichandran Iyer Jul 31 '17 at 13:38
  • Exactly. The same function can be linear or not depending on the base field. For instance, complex conjugation is linear when you consider $\Bbb C$ as a two-dimensional real vector space, but not at all when considered as a function between one-dimensional $\Bbb C$-vector spaces. However, it is additive either way. – Arthur Jul 31 '17 at 13:40
  • That's a better example of the phenomena. Thank you for all these clarifications. I am sure it will help the questioner as well. – Sarvesh Ravichandran Iyer Jul 31 '17 at 13:42
  • I dont understand how you are pulling out vectors e2, e2 out and dividing them. What is the meaning of e2/e1? Don't you mean( Something like) $f(e_2)=f(\frac{e_2}{f(e_1)})=$? – MSIS Aug 06 '20 at 04:47
  • @MSIS They are elements of a basis of $\mathbb R$ over $\mathbb Q$, so $e_2$ and $e_1$ are elements of $\mathbb R$ (and non-zero at that, because $0$ cannot belong in a basis). That is why I can divide one by the other. – Sarvesh Ravichandran Iyer Aug 06 '20 at 04:48
  • Apologies for my dumb comment, I somehow assumed ( for no good reason) that it was a general argument on vector spaces where the ei were basis elements. – MSIS Aug 06 '20 at 17:22
  • @MSIS No problem, I would have expected that confusion, so thanks for asking! – Sarvesh Ravichandran Iyer Aug 06 '20 at 17:46
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Such a function cannot be defined explicitly. It only can be shown (assuming the axiom of choice), that such a function exists. First note that each such function $T:\mathbb R\to\mathbb R$ is a solution to the so called Cauchy functional equation. It can be shown that $T$ is automatically linear, if $T$ is Lebesgue-measurable or continuous or bounded. In particular, non-linear examples have to be really pathological.

For a more detailed discussion and a proof of the existence of a nonlinear example see

https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation

leftaroundabout
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sranthrop
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