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A family of open subsets is a subbasis of a topology when its finite intersections form a basis of a topology, including the empty intersection equal to the entire space. For the product topology, inverse images of open sets under the product projections form the standard subbasis of a topology. A map into the space is continuous if the inverse image of every subbasic set is open.

For the normalized discrete Fourier transform basis over ${\mathbb{Z}}_N$, products of coefficient functions satisfy $f_m{f}_n=N^{-1/2}{f}_{m+n}$. Taking adjoints of the multiplication rule for the basis-copying isometry gives the displayed identity, with addition modulo $N$. Its $N$ output terms are orthonormal, so the prefactor is required by norm preservation. Each reduced quantum state is $I/N$, making the output a maximally entangled state for $N\ge 2$. This explains concretely why copying one basis is incompatible with copying its Fourier basis.

The unique scalar coefficients in the finite linear combination expressing a vector in a chosen basis. Spanning gives existence, while linear independence makes the difference of two proposed representations have all zero coefficients. Ordering a finite basis turns these coefficients into a coordinate vector.

Pressure changes the melting temperature of ice according to the Clapeyron equation. Ice pushed against an obstacle can melt on its high-pressure side and refreeze downstream where pressure is lower. Conduction of the released latent heat back towards the melting region permits motion around the obstacle. Small obstacles favor this heat-transfer mechanism; larger ones can be bypassed mainly by ice deformation.

A water-filled cavity opens where the normal contact pressure behind a bed obstacle would otherwise fall below subglacial water pressure. Reduced effective pressure in subglacial drainage enlarges cavities and changes the resisting contact geometry. For a fixed bounded-slope bed, the basal shear stress is bounded by a bed-dependent multiple of effective pressure; a sliding curve may saturate or weaken as cavities grow.

For a shallow glacier with basal traction proportional to thickness and sliding law $u_b\propto {\tau }_b^m$, negligible internal-deformation flux gives $u_b\propto h^m$. After scaling, the thickness flux is $h^{m+1}$. Linearization around a steady thickness $h_{\ast }(x)$ gives ${\zeta }_t+[c(x)\zeta {]}_x=0$, $c=(m+1)h_{\ast }^m$. Thickness disturbances therefore propagate faster than the surface material speed $u_s\simeq h_{\ast }^m$ by factor $m+1$.

A steady accumulation profile and upstream discharge $q_{\mathrm{in}}$ give $h_{\ast }^{m+1}=q_{\mathrm{in}}+{\int }_0^{x}a(\xi )d\xi$. Along a disturbance ray, $dx/dt=c$ and $d\zeta /dt=-c^{\prime }\zeta$, so a spatially varying background also changes amplitude. The surface-speed ratio is specific to sliding-dominated flow; internal deformation and significant surface-slope corrections change the flux derivative.

A normed vector space is a Banach space exactly when every absolutely convergent series converges in the space. Completeness implies the assertion because the triangle inequality makes the partial sums a Cauchy sequence. Conversely, select from any Cauchy sequence a subsequence whose consecutive differences have norm at most $2^{-k}$. The series of these differences converges by hypothesis, so the subsequence converges. The original Cauchy sequence then has the same limit. This provides a useful way to demonstrate incompleteness by a summable-norm series whose candidate sum lies outside the space.

Suppose a finite perfect-information game has no chance moves, and every player strictly ranks the possible outcome labels. Different terminal histories may have the same outcome label and thus the same payoff. Every subgame then has a unique subgame perfect equilibrium outcome, even though its equilibrium strategy need not be unique.

The proof is backward induction. Each child subgame has a unique continuation outcome. The moving player's strict ranking selects exactly one best label among those outcomes. Several actions can tie only by leading to that same label, which proves uniqueness at the current node. For a sequential election, the labels may be winning candidates: a voter can have several optimal ballots, all producing its same favourite achievable winner. If preferences have ties between different labels, the conclusion can fail.

For an independent axiom set, semantic entailment of any one axiom from the remaining axioms fails. By the completeness theorem, this is equivalent to no member being formally derivable from the others in a complete calculus. To establish independence it suffices to give, for each axiom, a first-order structure satisfying all the other axioms and failing that one.

An equivalence-relation first-order structure with exactly two infinite classes, one countable and one uncountable, is atomic but lacks finite-tuple homogeneity. Its first-order theory has quantifier elimination: a new point is either assigned a previously used class or an available unused class, and can avoid finitely many forbidden points in that infinite class. Finite equality-and-equivalence patterns therefore isolate all tuple partial types. All elements have the same complete one-type, but no automorphism exchanges the two differently sized classes.