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368 votes
31 answers
80k views

Geometric interpretation of trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandary; Is there a geometric interpretation of the trace of a matrix? This question ...
152 votes
18 answers
24k views

Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
128 votes
13 answers
27k views

Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
Frank Thorne's user avatar
  • 7,347
127 votes
4 answers
32k views

Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is: A vector space has the same dimension as its dual if and only if it is finite dimensional. I have seen a total of one proof of ...
115 votes
3 answers
5k views

The number $\pi$ and summation by $SL(2,\mathbb Z)$

Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality) Then, we discovered by heuristic arguments and then verified by computer that $$\...
Nikita Kalinin's user avatar
109 votes
15 answers
12k views

Why are matrices ubiquitous but hypermatrices rare?

I am puzzled by the amazing utility and therefore ubiquity of two-dimensional matrices in comparison to the relative paucity of multidimensional arrays of numbers, hypermatrices. Of course ...
Joseph O'Rourke's user avatar
109 votes
19 answers
38k views

Why were matrix determinants once such a big deal?

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
94 votes
1 answer
11k views

The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ...
algori's user avatar
  • 23.5k
91 votes
19 answers
20k views

Injectivity implies surjectivity

In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example, Set theory An injective map between two finite sets with the same cardinality is surjective. ...
91 votes
5 answers
124k views

Eigenvalues of matrix sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite? I am ...
Jean-Pierre Gunman's user avatar
86 votes
1 answer
6k views

Are there non-scalar endomorphisms of the functor $V\mapsto V^{**}/V$?

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor $$ V\mapsto V^{**}/V $$ of the category of $K$-vector spaces? I asked a related question on Mathematics Stackexchange, but ...
Pierre-Yves Gaillard's user avatar
81 votes
4 answers
8k views

Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I: The theory of commutative ...
Jonas Meyer's user avatar
  • 7,329
81 votes
10 answers
9k views

Existence of a zero-sum subset

Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes: Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...
Gjergji Zaimi's user avatar
81 votes
3 answers
9k views

Norms of commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
Bill Johnson's user avatar
  • 31.5k
77 votes
0 answers
4k views

2, 3, and 4 (a possible fixed point result ?)

The question below is related to the classical Browder-Goehde-Kirk fixed point theorem. Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$ be a mapping such that $$\Vert Tx-Ty\...
Ady's user avatar
  • 4,060
71 votes
16 answers
21k views

Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.] I've read looked at the examples in most ...
71 votes
2 answers
6k views

Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?

I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, ...
Andrew Stacey's user avatar
69 votes
3 answers
12k views

Nonconvexity and discretization

Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem. Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
Peter Scholze's user avatar
68 votes
4 answers
9k views

explicit big linearly independent sets

In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or ...
Martin Brandenburg's user avatar
66 votes
7 answers
10k views

Why is the Hahn-Banach theorem so important?

Every time I hear it mentioned it is praised in the highest possible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only ...
teil's user avatar
  • 4,351
66 votes
2 answers
8k views

Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
Per Vognsen's user avatar
  • 2,071
66 votes
3 answers
4k views

Does linearization of categories reflect isomorphism?

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
Tilman's user avatar
  • 6,162
65 votes
9 answers
12k views

Polish spaces in probability

Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed. Question: What can go wrong when doing probability on non-Polish spaces?
Thanh's user avatar
  • 651
65 votes
14 answers
6k views

Notions of convergence not corresponding to topologies

This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago: Exam question: Is there a metric on the ...
jon's user avatar
  • 801
63 votes
5 answers
10k views

Jean Bourgain's relatively lesser known significant contributions

Jean Bourgain passed away on December 22, 2018. A great mathematician is no longer with us. Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions,...
63 votes
7 answers
9k views

How to prove this determinant is positive?

Given matrices $$A_i= \biggl(\begin{matrix} 0 & B_i \\ B_i^T & 0 \end{matrix} \biggr)$$ where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following? $$\det \big( I + e^...
Lei Wang's user avatar
  • 845
62 votes
25 answers
70k views

Linear Algebra Texts?

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
62 votes
9 answers
23k views

Can a vector space over an infinite field be a finite union of proper subspaces?

Can a (possibly infinite-dimensional) vector space ever be a finite union of proper subspaces? If the ground field is finite, then any finite-dimensional vector space is finite as a set, so there are ...
Anton Geraschenko's user avatar
61 votes
11 answers
11k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
Daniel Litt's user avatar
60 votes
23 answers
108k views

A good book of functional analysis [closed]

I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics)....
59 votes
9 answers
10k views

Motivation for and history of pseudo-differential operators

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
shuhalo's user avatar
  • 5,327
59 votes
7 answers
29k views

Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
57 votes
6 answers
6k views

Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
Konrad Swanepoel's user avatar
56 votes
21 answers
18k views

Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
56 votes
21 answers
14k views

Linear algebra proofs in combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
55 votes
0 answers
2k views

What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?

Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
Dmitri Zaitsev's user avatar
54 votes
5 answers
2k views

Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra

Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge ...
Dylan Thurston's user avatar
53 votes
7 answers
51k views

Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that $$\det(A+B) \ge \det(A) + \det(B)$$ in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-...
user15221's user avatar
  • 541
53 votes
9 answers
13k views

Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
agt's user avatar
  • 4,306
53 votes
3 answers
13k views

Pullback measures

Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist? It's true that the naive treatment of such a ...
Alex M.'s user avatar
  • 5,407
53 votes
5 answers
5k views

Does this formula have a rigorous meaning, or is it merely formal?

I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer ...
Dick Palais's user avatar
  • 15.3k
52 votes
5 answers
4k views

When exactly and why did matrix multiplication become a part of the undergraduate curriculum?

The story about Heisenberg inventing matrices and matrix multiplication in 1925 is very well known and well documented. A few weeks later, Born and Jordan read this work and recognized matrix ...
Alexandre Eremenko's user avatar
52 votes
2 answers
3k views

vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct. This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ...
Nik Weaver's user avatar
  • 42.8k
51 votes
22 answers
19k views

Why linear algebra is fun!(or ?)

Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor. I'm doing an introductory talk on linear algebra with the ...
51 votes
2 answers
5k views

A strengthening of the Cauchy-Schwarz inequality

Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
Nathaniel Johnston's user avatar
50 votes
7 answers
16k views

Way to memorize relations between the Sobolev spaces?

Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel ...
Orbicular's user avatar
  • 2,935
49 votes
14 answers
21k views

Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
49 votes
3 answers
4k views

Is this proof of Perron's theorem correct, and if so is it original?

A few years ago, I came up with this proof of Perron's theorem for a class presentation: https://pi.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf I've written an outline of it below ...
Hannah Cairns's user avatar
48 votes
6 answers
7k views

Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "...
47 votes
3 answers
3k views

Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as $$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$ is a harmonic function, that is $\Delta V = 0$ where ...
Sandeep Silwal's user avatar

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