My professor listed six normed function spaces on the board and asked, "Which are complete? For those that are not complete, what is their completion?" I'm wondering if there is a fast way to intuit the answers to these questions? Or does one just build a stockpile of examples over time?
One technique I think I've figured out: If normed space $X, \| \cdot \|$ is a subspace of Banach space $Y, \| \cdot \|$ then $X$ is complete if and only if $X$ is closed in the topology of $Y$. For example, $C(K), \| \cdot\|_\infty$ for compact $K$ is (isomorphic to) a closed subspace of $L^\infty(K)$, but $C_0(K), \| \cdot \|$ is (isomorphic to) a subspace of $L^\infty(K)$ that is not closed. So in this context, completeness boils down to close-ness. Are there other such quick ways to mentally intuit whether or not a space is complete?
And what about the second question? Is there a way to quickly know (or at least guess) what the completion of a space is?
This is absolutely an issue of having a repertoire/stockpile of the iconic examples!!! :)
Many of these are not so easy to figure out "on the spot"... but, still, do illustrate important and iconic principles.
One of those iconic things is about the closure of $C^o_c(\mathbb R)$, compactly-supported continuous (real-or-complex-valued) functions on $\mathbb R$, in the sup-norm topology, inside the space $C^o_{\mathrm {bdd}}(\mathbb R)$ of bounded continuous functions with sup norm. It's the space of "continuous functions going to $0$ at infinity"... EDIT: an important phrase here is that "supports can leak out to infinity" in a sup-norm Cauchy sequence of compactly-supported continuous functions.
But/and, again, this is not obvious... except in hindsight. So, surely, the reasonable sense of those questions your instructor asked is that you should do the work to develop/refine your intuition, to incorporate the facts (as you will shortly determine... :)