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Browsing through stackexchange, I'm often struck by the elegance of topological/algebraic proofs for claims in analysis. Time and again, I can come up with one proof but would not have thought of the other, with the ones I think of coming from an analyst's, or sometimes more sacrilegiously, an engineer's perspective. I'm good at juggling around epsilons, establishing inequalities, or proving one of many types of convergence, but whenever the key step is to invent new spaces as simple as a quotient space, I get stuck. Infuriatingly, I tend to understand the proof once it is presented - I just cannot come up with it on my own. Let me illustrate with an example:

The claim to be proven is the following:

Let $T: X \rightarrow Y $ be a linear mapping between two normed linear spaces. Assume you have proved that if $X$ is finite dimensional, $T$ is continuous. Prove that if $Y$ is finite dimensional, $T$ is continuous if and only if $\ker(T)$ is closed.

If $T$ is continuous, then since $\ker(T) = T^{-1}(0)$ is the preimage of a closed set, it is closed. So far so good. For the second portion I have two different proofs:

The first proof is gritty, and in a style of an analysis textbook:

Without loss of generality, $T$ is surjective. Let $\{e_i\}_{i=1}^n$ be a basis for $Y$. Then there exists $\{u_i\}_{i=1}^n$ such that $Tu_i = e_i$. If $T$ were not continuous, there is a sequence $\{x_j\}_{j=1}^\infty \rightarrow 0$ such that $\|Tx_j\|=1$. The unit sphere in $X$ in $Y$ is compact, hence there is a subsequence (for simplicity just $\{x_j\}_{j=1}^\infty$) and a $y \in Y$ with $\|y\| =1$ such that $Tx_j \rightarrow y$. Since $Y$ is finite-dimensional, we have for some appropriate coefficients that $$Tx_j = \sum_{i=1}^n \alpha_{i,j} e_i, \quad y = \sum_{i=1}^n \alpha_i e_i , \text{ where } \alpha_{i,j} \rightarrow \alpha_i \text{ as } j \rightarrow \infty .$$ Denote $w_j = \sum_{i=1}^n \alpha_{i,j} u_i, w = \sum_{i=1}^n \alpha_i u_i $. Note that $T(w_j - x_j) = 0$, so $w_j - x_j \in \ker(T)$. Since $\ker(T)$ is closed $w = \lim w_j - x_j \in \ker(T)$, but $Tw = y \neq 0$, a contradiction. Hence $T$ must be continuous.

The second proof is somewhat topological, and much more clean and elegant:

Since $\ker(T)$ is closed, $X/\ker(T)$ is a normed linear space. Define $\bar{T}:X/\ker(T) \rightarrow Y$ by $\bar{T}(x + \ker(T)) = T(x)$, which is linear, and continuous since $X/ker(T)$ is finite-dimensional. Define $\pi: X \rightarrow X/\ker(T)$ in the obvious way. Then note that $T = \bar{T} \circ \pi$ is a composition of continuous function, hence continuous.

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This is definitely not a perfect example, but in my imperfect vocabulary, the first, gritty proof is very detailed, full of analysis concepts such as convergence, or simple linear algebra, whereas the second deals with properties of the space as a whole, without relying much on the objects within it. I realize that it is hard to compare these 'styles' in proof, and the first style can go a long way. Nevertheless, I feel that to become a better student of mathematics, and even of analysis, I should get a better handle on algebraic or topological techniques. I've taken courses in abstract algebra and topology, but my intuition for these has not grown sufficiently for me to think of questions in analysis in largely algebraic or topological ways, at least beyond standard linear algebra.

In the end, I am sure that I want to study analysis, mostly for applied purposes, and I'm aware that there are practical constraints - I won't be in school forever. Would you still agree that developing greater topological intuition is worthwhile? Why or why not? If you feel that it is worth it, how should I go about cultivating said intuition (keeping in mind that eventually, I will enter applied rather than pure math)?

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The question is one one hand understandable. On the other hand, I do not believe my attempt as answer to be considered anywhere near "sufficient". Several points of view should be taken into account, and I have focused on examples. These are examples have helped me "read as a mathematician in my opinion would". That being said, my main rule of thumb is: if you can draw a strategy, you are probably making things easy for yourself.

1) Constructions in mathematics are often designed with specific goals in mind. One example has already described by you: quotients. From my point of view, quotients are abstract constructions given by certain ingredients: some set/space/algebraic object $A$ or likewise, some subset/subspace etc. $B$ and the quotient map $\pi \colon A \rightarrow A/B$, with $A/B$ all equivalence classes.

If we for instance just consider the topological setting, I would wish my new space $A/B$ to have a topology as well. Further, it seems natural to force $\pi$ to be continuous (being part of the construction). $\textit{In fact, the quotient topology is the topology making this happen!}$

2) With these general ideas behind constructions, I often find myself better equipped at rephrasing a problem, such as yours above, into a specific question concerning an object. In the "clean" proof you mentioned, I consider the approach as follows: Since compositions of continuous maps remain continuous, it suffices to write $T$ as a product (composition) of continuous maps. From this point one, we just need to rewrite $T$ as $$ T\colon X \stackrel{\pi}{\rightarrow} V \stackrel{\varphi}{\rightarrow} Y $$

For some normed space $V$. Such a diagram is typically called a factorization without much of a surprise. The map $\bar{T}$ you defined is the unique map $\varphi$ on the quotient space $V=X/\ker T$ factorizing $T$ in this manner (with $V=X/\ker T$ and $\pi$ the quotient map).

3) To summarize a bit, good tips are the following.

  • Always try to reduce your problem to something managable/simple such as writing $T$ as a composition of maps. Algebraic properties often help you here. An easy but good example would be a basis. Say you need to prove the existence of some linear map, then achieving this on a prescribed basis suffices and you may forget about the rest of some possibly crazy space. Again, this is natural, for one of the points of a basis is to reduce the "size".

  • Draw your problem if possible, using diagrams or exact sequences. Why? Constructions, especially in algebra (in my opinion), are often build with such diagrams or sequences describing them. Occasionally, you end up drawing a diagram which heavily resembles the one attached to a construction (for inspiration or perhaps direct hints).

  • Know the ideas of the constructions. My favourite example is the first isomorphism theorem for modules. Namely, if $\varphi \colon A \rightarrow B$ is a surjection module-map, then $$ A/\ker \varphi \cong B. $$

True there is such an isomorphism. But the map itself is important (and pops up in your example!): each such, not necessarily surjective, $\varphi$ induces a module map $\bar{\varphi} \colon A/\ker \varphi \rightarrow B$ and the first isomorphism states that surjectivity of $\varphi$ implies that $\bar{\varphi}$ is an isomorphism. Ergo you always have that associated map at your disposal and it factorizes $\varphi$ as \begin{equation}\tag{$\ast$} \varphi\colon A\rightarrow A/\ker \varphi \rightarrow B. \end{equation}

where the first arrow represents the quotient map. (If you ask me, the existence of $\bar{\varphi}$ is the crucial part of proving the first isomorphism theorem, not why one acquires an isomorphism whenever $\varphi$ is surjective).

If we return to your example: Step $1$ was to reduce the problem into writing $T = \bar{T} \circ \pi$. Step $2$, from my point of view, is to rephrase the problem into finding a diagram $$ T\colon X \stackrel{\pi}{\rightarrow} V \stackrel{\varphi}{\rightarrow} Y $$

However, this diagram appears quite often in quotients as in ($\ast$). Exploiting this might work?

I hope this helps.

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