# Can we write limits variable-free?

When first learning the subject and when doing simple calculations, it's convenient to describe derivatives in terms of variables, i.e. $$\frac{d}{dx} f(x) = f'(x)$$ and we say the derivative takes a "function" (expression) and maps it to another "function" (expression). But more rigorously, we can say that given an $$n$$-manifold $$M$$, a chart $$x:U\subseteq M\rightarrow\mathbb{R}^n$$, and a function $$f:M\rightarrow \mathbb{R}$$, $$\frac{\partial}{\partial x_i} f \equiv \partial_i(f\circ x^{-1})\circ x$$ This is just one way of reformalizing derivatives, but what we have done is taken out the reliance on a naive notion of "variables". I was wondering if we have made any similar re-formulations for the limit. From the above equivalence, we need the notion of a limit to fully describe the partial derivative on $$\mathbb{R}^n$$. I.e. for a function $$g:\mathbb{R}^n\rightarrow\mathbb{R}$$, $$\partial_ig(a_1,...,a_i,...,a_n)\equiv \lim_{h\to 0} \frac{g(a_1,...,a_i-h,...,a_n)-g(a_1,...,a_i,...,a_n)}{h}$$ You could use an "epsilon-delta" definition of limits if $$n=1$$ and you want to totally-order $$\mathbb{R}$$ or you could define limits using neighborhoods or open sets, but the notation is what I'm stuck on. Is there a way to formalize the limit as a map $$\lim_{i,\ b}:\ C^0(\mathbb{R}^n)\ \to\ C^0(\mathbb{R}^n)$$ $$\left(\lim_{i,\ b}g\right)(a_1,...,a_i,...,a_n)\equiv\ \,\lim_{a_i\to b}\left(g(a_1,...,a_i,...,a_n)\right)"$$ similar to the many ways we have reformatted the derivative? For clarity, my question is this:

Question: Is there a definition of the limit on Euclidean $$\mathbb{R}^n$$ space that doesn't require variables? If not, is there simply a notion of a limit that doesn't use variables in its notation?

• How familiar are you with the topological definitions of limits, particularly the usage of nets and such? That seems at least along the lines of what you're looking for... Apr 19, 2020 at 19:29
• @StevenStadnicki I'm familiar with the definition of limit of a function $f:X\to Y$ for topological spaces $X$ and $Y$... For every open set $U$ of $X$ containing $p$, there's an open set $V$ of $Y$ such that the image of $U$ under $f$ is a subset of $V$. I'm pretty sure that's the definition I remember. I see that generalization to nets uses a variable-free notation, but how then would you notate the limit of the difference quotient? The traditional notation allows for "embedded" variables, but I can't find a way that represents this in a variable-free notation. Apr 19, 2020 at 20:08
• "that doesn't require variables": what do you mean ? Even in the notation of derivatives there are variables.
– user65203
Apr 20, 2020 at 9:23
• @YvesDaoust I mean exactly what I say and the notation for derivatives in fact does not use variables; e.g. $\frac{\partial}{\partial x_i}$ only includes the symbols $\partial$, $x$, and $i$; $\partial$ is simply a symbol that helps us conceptually but it's certainly not a variable, $x$ is a chart which is a specific element of the atlas of the manifold so that's not a variable either, and $i$ is a specific integer between $1$ and $n$ so that's not a variable either. Apr 20, 2020 at 18:10
• "$x$ is a chart which is a specific element of the atlas of the manifold so that's not a variable ": wow, I am afraid I have nothing to do here.
– user65203
Apr 20, 2020 at 18:22

A friend of mine, Jakub Marian, developed a coordinate free notation in calculus in his bachelor thesis: Alternative mathematical notation and its applications in calculus He was pretty good so there should be very good ideas.

I am looking at it now and he only treats calculus of 1 variable! You can do the case of $$n$$-variables yourself :-)

A crucial role is played by the identity function $$\iota(x) = x$$ which he uses to write compositions efficiently; e.g., the simple expression $$(\iota^3+\iota^2 + \iota)\circ(\iota^3 + 2\iota)\circ(\iota^2 + 8\iota + 4\iota^2)$$ in his notation corresponds to a crazy polynomial in the variable $$x$$. He views $$\lim_a$$ as an operator on functions (he uses the "undefined" symbol $$\Omega$$ if the limit does not exist). He has the "difference quotient with step $$h$$" operator $$\Delta_h$$ and defines "indefinite sums with step $$h$$" as $$\sum_h = (\Delta_h)^{-1}$$ (defined up to the set of $$h$$-periodic functions). Based on this, he develops a theory of the derivative $$\partial$$ and the integration $$\int$$ (in the sense of an inverse to $$\partial$$) and rewrites the classical theorems in this notation. He illustrates at many places how his notation simplifies some computations.

However, I can not imagine any real application of it in the sense of gathering more information about spaces of functions, etc. On the other hand, I was once thinking of using similar abstractions when I was supposed to compute some crazy integrals in my research and had the suspicion that there is a hidden structure which I just didn't see when looking at the crazy coordinate expressions. Such notation is also better if you communicate with a machine; e.g., from my experience, Mathematica and Haskell think about expressions in this way. It is probably very much related to $$\lambda$$-calculus but I am not an expert on that.

• This is interesting, but I did not see the definition of the notation $\Delta_\bullet$ (used in Definition 2.19). The definition $\partial = \lim_0 \Delta_h$ made sense, but it still has the variable $h$, while think the idea of this question is to eliminate $h$. Apr 19, 2020 at 22:06
• He defines it in Definitions 2.10 and 2.11 (limit of a family of functions and a family of operators). It would be probably more elegant at this point to have the generalization for n variables. Apr 19, 2020 at 22:46
• So the answer is actually that $\Delta_\bullet$ is not defined as an independent notation, the way $\Delta_h$ is; instead it is defined only in conjunction with the $\lim$ symbol. It seems a little strained, though the earlier definitions such as $\lim_a$ seemed intuitive enough. Apr 19, 2020 at 23:42
• I would understand $\Delta_\bullet$ independenly as a family of operators $\Delta_{h}$ for $h\in \mathbb{R}$, i.e., a particular set. He defines the limit of a family of operators, e.g., $\lim_0 \Delta_\bullet$, as the operator associating to a function $f$ the function which is the limit of the family of functions $\Delta_h f$. The limit of a family of functions $g_h$ is then defined pointwise like $(\lim_0 g_h)(x) := \lim_0 (g_h(x))$. Apr 20, 2020 at 8:29
• Correction of the last sentence: The limit of a family of functions $g_\bullet$ is then defined pointwise as the function $(\lim_0 g_{\bullet})(x) = \lim_0 (g_{\bullet}(x))$. Again, I see a possibility of making this better by developing the formalism for more variables so that $\Delta_\bullet$ can be interpreted as an operator from functions of $1$ variable to functions of $2$ variables. Apr 20, 2020 at 9:51