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Cantor space is the binary sequence space with the product topology of discrete two-point factors. Here the Bernoulli-space synonym refers to this topology, without selecting a particular probability measure. The metric $d(x,y)={\sum }_{j\ge 1}{2}^{-j}|x_j-y_j|$ induces the topology. A diagonal subsequence argument gives compactness. Fixing a finite prefix gives a clopen cylinder set. The map $x\mapsto {\sum }_{j}2x_j{3}^{-j}$ is a homeomorphism to the usual Cantor set.

The smallest closed convex set containing a set $C$, equivalently the closure of its convex hull in the specified topology. The topology must be stated: the norm topology, weak topology and weak-star topology need not give the same closure for an arbitrary set.

Two nontrivial absolute values on a field are equivalent when they induce the same topology. Equivalently, one is a positive real power of the other.

A function space is a collection of functions with common domain and codomain, often supplied with a vector space structure and a norm or topology. Examples include L2 spaces and Sobolev spaces.

A sequence of functionals Gamma-converges in a specified topology when every convergent $u_n\to u$ satisfies $F(u)\le {\mathrm{liminf}}_n{F}_n(u_n)$, and each $u$ has a recovery sequence $u_n\to u$ with $F_n(u_n)\to F(u)$. With appropriate compactness of energy sublevels, limits of approximate global minimizers minimize $F$, and the minimum values converge. This is suited to variational approximation such as the Ambrosio–Tortorelli approximation; it does not describe arbitrary local minima.

A nonempty closed subset which, with its induced topology, is an irreducible topological space. In an affine algebraic set these subsets correspond, with reversed inclusion, to prime ideals of the coordinate ring.

For Fréchet spaces, the two continuity conditions for a linear map are equivalent. Original continuity makes every target continuous linear functional compose to a source continuous linear functional, proving weak continuity. Conversely, weak continuity makes these compositions continuous in the original source topology. They separate target points, so limits in the graph of a linear operator remain in the graph. The closed graph theorem for Fréchet spaces finishes the proof.

Without the hypotheses, equal continuous dual spaces do not imply equal topologies. The identity from an infinite-dimensional Hilbert space with its weak topology to its norm topology is weak-to-weak continuous but not continuous in those original topologies.

For objectives $D+\alpha J$ with $J\ge 0$ and positive parameter, suppose nearby-parameter global minimizers have uniformly bounded penalty, stay in a bounded set with convergent subsequences, and $D$ and $J$ are sequentially lower semicontinuous in the chosen topology. Their optimality inequalities and lower semicontinuity show that every subsequential limit minimizes the limiting objective. Uniqueness makes the regularized solution continuous in the parameter. Existence alone does not make an arbitrary selection among multiple global minimizers continuous.