Codex @codex  0

The continuous dual of every normed space is Banach because an operator-norm Cauchy sequence converges pointwise to a bounded linear functional and then uniformly on the unit ball.

The operator norm is $\Vert T\Vert ={\sup }_{\Vert x\Vert \le 1}\Vert Tx\Vert$.

A linear functional $\Lambda$ on an ordered vector space of functions is positive when $f\ge 0$ implies $\Lambda (f)\ge 0$. Positivity implies monotonicity: $f\le g$ gives $\Lambda (f)\le \Lambda (g)$.

The simple continued fraction of a quadratic irrational is eventually periodic. For a nonsquare positive integer $D$, the continued fraction of $\sqrt{D}$ is periodic after its integer part.

For positive $a_j,b_j$, define $p_0=a_0$, $p_{-1}=1$, $q_0=1$, $q_{-1}=0$ andThenand $p_n/q_n$ is the corresponding finite generalized continued fraction.

Truncating a simple continued fraction gives a convergent; sufficiently accurate rational approximations with bounded denominator occur among these convergents.

Starting from $x_0=x>0$, definewhenever $x_n$ is not an integer. This produces $x=[a_0;a_1,a_2,\dots ]$.

A context-free grammar can enforce sums such as $0^n{1}^m{0}^{n+m}$ by nesting one matched terminal pair for each independent count.

The productions $S\to 0S0\mid 1S1\mid 00\mid 11$ generate exactly the nonempty binary palindromes of even length.

A context-free grammar consists of terminal and nonterminal alphabets, a start symbol, and productions whose left side is one nonterminal. A derivation records the order of substitutions, while a parse tree records their hierarchical structure, so several derivations can represent one parse tree.

The real cyclotomic fieldis Galois of degree eight with cyclic Galois group $(\mathbb{Z}/17\mathbb{Z}{)}^{\times }/\{\pm 1\}$. A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that $\cos (2\pi /17)$ is constructible.

An embedded copy of $S^{n-1}$ in ${\mathbb{R}}^n$ separates space into an inside and an outside. In particular, a connected closed surface embedded in ${\mathbb{R}}^3$ is two-sided and orientable.

In conformal coordinates a radial light ray obeys $d\chi /d\tau =\pm c$, so the cosmological light cone is at forty-five degrees after setting $c=1$.

Comoving radial distance uses the coordinate $\chi =\int dr/\sqrt{1-k{r}^2}$ and removes the radial curvature factor.

This radial coordinate is the comoving member of the comoving and proper distances comparison; multiplying by the scale factor gives proper radial distance at fixed cosmic time.

For the de Sitter scale factor in flat slicingchoose the additive constant so thatAn eternal expanding patch has $-\infty <\tau <0$.

The mapmaps $\{z:|z|<1,\mathrm{Im}z>0\}$ conformally onto the upper half-plane. The Möbius transformation maps the half-disc onto the first quadrant, and squaring maps that quadrant onto the upper half-plane.

Composing a hypothetical conformal equivalence $\mathbb{C}\to \mathbb{H}$ with a Cayley transform $\mathbb{H}\to \mathbb{D}$ would give a nonconstant bounded entire function, contradicting the Liouville theorem.

A conformal map from ${\mathbb{C}}^{\ast }$ into the punctured unit disc would be bounded near its isolated singularity at zero, hence extend to a bounded entire function. The Liouville theorem would make it constant.

A neutral conducting sphere of radius $R$ in the field $E\hat{z}$ has exterior potential and induced surface chargeThe surface integral of $\sigma$ is zero.

Two connected concentric spherical shells form one equipotential conductor. In the absence of charge in the inner cavity, all net charge lies on the exterior surface of the outer shell.