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For vector spaces, $\dim (U + V) = \dim U + \dim V - \dim (U \cap V)$, so $$ \dim(U +V + W) = \dim U + \dim V + \dim W - \dim (U \cap V) - \dim (U \cap W) - \dim (V \cap W) + \dim(U \cap V \cap W), $$ …

It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a que …

A proof of the identity $$1+2+\cdots + (n-1) = \binom{n}{2}$$ (Adapted from an entry I saw at Wolfram Demonstrations, see also the original faster animation) This proof was discovered by Loren Lar …

Irrationality of $2^{1/n}$ for $n\geq 3$: if $2^{1/n}=p/q$ then $p^n = q^n+q^n$, contradicting Fermat's Last Theorem. Unfortunately FLT is not strong enough to prove $\sqrt{2}$ irrational. I've forg …

Everyone knows that for any two square matrices $A$ and $B$ (with coefficients in a commutative ring) that $$\operatorname{tr}(AB) = \operatorname{tr}(BA).$$ I once thought that this implied (via ind …

"If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." Althought the statement sounds obvious, it is actually independent of ZFC. The statement follows from the …

The closure of the open ball of radius $r$ in a metric space, is the closed ball of radius $r$ in that metric space. In a somewhat related spirit: the boundary of a subset of (say) Euclidean space ha …

The Busemann-Petty problem (posed in 1956) has an interesting history. It asks the following question: if $K$ and $L$ are two origin-symmetric convex bodies in $\mathbb{R}^n$ such that the volume of e …

Many students believe that 1 plus the product of the first $n$ primes is always a prime number. They have misunderstood the contradiction in Euclid's proof that there are infinitely many primes. (By …

I don't even know if this is intentional or not. In his book Teichmuller theory, John Hubbard frequently references the category of Banach Analytic Manifolds. He adheres to the convention that a cat …

Mathematical notation in a given mathematical field $X$ is basically a correspondence $$ \mathrm{Notation}: \{ \hbox{well-formed expressions}\} \to \{ \hbox{abstract objects in } X \}$$ between mathem …

An example that came up in my measure theory class today: The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges, because otherwise the functions $f_n := \frac{1}{n} 1_{[0,n]}$ would be dominat …

Topology is the art of reasoning about imprecise measurements, in a sense I'll try to make precise. In a perfect world you could imagine rulers that measure lengths exactly. If you wanted to prove th …

Here's my list of false beliefs 😉 If $U$ is a subspace of a Banach space $V$, then $U$ is a direct summand of $V$. If $M/L$ and $L/K$ are normal field extensions, then the same is true for $M/K$. Su …

I don't know if this is common or not, but I spent a very long time believing that a group $G$ with a normal subgroup $N$ is always a semidirect product of $N$ and $G/N$. I don't think I was ever sho …