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It seems like you're thinking about adjoints with respect to an inner product, but I find it more natural to think of adjoints as a way to get a map $B^{\ast} \to A^{\ast}$ from a map $A \to B$ and th …

The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; …

I'll assume we're talking about complex functions; if real, tensor with $\mathbb{C}$. Now pass to the group of units. With the topology given by spectral radius (this is an algebraic description of th …

Unitization and metric completion are both left adjoint functors, as are may other "-tion" operations in mathematics, such as localization or abelianization. Specifically, there is a forgetful functor …

It's a standard series computation to show that $$ \sum_{n \ge 1} \frac{x^n}{n} = \log \frac{1}{1 - x} $$ Now substitute $x = e^{i t}$ and take the real part. (As an aside, the reason I write th …

The spectral theorem in the finite-dimensional case is important in spectral graph theory: the adjacency matrix and Laplacian of an undirected graph are both symmetric, hence both have real eigenvalue …

I can answer your last question, at least. The derivative acts as a shift operator on Taylor series, so the operator $\frac{d}{dx} - 1$ acts as the forward difference on Taylor series. So their eige …

As far as I can tell, every example except the last is representation theory, understood in a suitably general sense. For example studying the eigenvalues of an operator is the same as studying the co …

Every monic integer polynomial $f(x) \in \mathbb{Z}[x]$ of degree $n$ is the characteristic polynomial of an $n \times n$ matrix, namely its companion matrix. The companion matrix is invertible iff th …

Already the desired result is false for $n = 3, m = 2$, but for simpler reasons than I suggested in the comments. In this case the polynomials you give are $x^2 - 2y^2 + z^2, xy$ and their permutation …

I do not believe that the Hahn-Banach theorem is necessary. At some point I had planned on writing up a blog post verifying this but I lost the motivation... The idea is that you can prove Liouville …

You want to take the category $\text{Ban}_1$ of Banach spaces and short maps (linear maps of operator norm $\le 1$). The unit ball functor $U : \text{Ban}_1 \to \text{Set}$ is represented by $\mathbb{ …

The Stone-Čech compactification of $\mathbb{R}$ is not its one-point compactification. The former is the largest compactification of a space, while the latter, if it exists, is the smallest compactifi …

I don't understand why you say that $f$ is continuous if its inputs are positive integers. Anyway, you can take any symmetric polynomial in the $g(r_i)$, e.g. $\sum_{i < j} g(r_i) g(r_j)$.

Your first requirement suggests to me that you want to think of an infinite-dimensional vector space as an ind-object, namely the filtered colimit of its finite-dimensional subspaces. If so, one forma …