In mathematics, the **Hilbert cube**, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).

The Hilbert cube is best defined as the topological product of the intervals [0, 1/*n*] for *n* = 1, 2, 3, 4, ... That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form the sequence .

The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval [0, 1]. In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension.

If a point in the Hilbert cube is specified by a sequence with , then a homeomorphism to the infinite dimensional unit cube is given by .

It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (i.e. a Hilbert space with a countably infinite Hilbert basis). For these purposes, it is best not to think of it as a product of copies of [0,1], but instead as

- [0,1] × [0,1/2] × [0,1/3] × ···;

as stated above, for topological properties, this makes no difference. That is, an element of the Hilbert cube is an infinite sequence

- (
*x*_{n})

that satisfies

- 0 ≤
*x*_{n}≤ 1/*n*.

Any such sequence belongs to the Hilbert space ℓ_{2}, so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the product topology in the above definition.

As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff theorem. The compactness of the Hilbert cube can also be proved without the Axiom of Choice by constructing a continuous function from the usual Cantor set onto the Hilbert cube.

In ℓ_{2}, no point has a compact neighbourhood (thus, ℓ_{2} is not locally compact). One might expect that all of the compact subsets of ℓ_{2} are finite-dimensional. The Hilbert cube shows that this is not the case. But the Hilbert cube fails to be a neighbourhood of any point *p* because its side becomes smaller and smaller in each dimension, so that an open ball around *p* of any fixed radius *e* > 0 must go outside the cube in some dimension.

Any infinite-dimensional convex compact subset of is homeomorphic to the Hilbert cube. The Hilbert cube is a convex set, whose span is the whole space, but whose interior is empty. This situation is impossible in finite dimensions. The tangent cone to the cube at the zero vector is the whole space.

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.

Every G_{δ}-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a separable and complete metric space. Conversely, every Polish space is homeomorphic to a G_{δ}-subset of the Hilbert cube.^{ [1] }

- ↑ Srivastava, pp. 55

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In topology and related areas of mathematics, a **metrizable space** is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis.

In topology and related branches of mathematics, a topological space is called **locally compact** if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

The **Baire category theorem** (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In mathematics, a **Cantor space**, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a **Cantor space** if it is homeomorphic to the Cantor set. In set theory, the topological space 2^{ω} is called "the" Cantor space.

In the mathematical discipline of general topology, a **Polish space** is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology and related branches of mathematics, **total-boundedness** is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size"

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In the mathematical field of point-set topology, a **continuum** is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. **Continuum theory** is the branch of topology devoted to the study of continua.

In functional analysis and related areas of mathematics, a **metrizable** topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

- Srivastava, Shashi Mohan (1998).
*A Course on Borel Sets*. Graduate Texts in Mathematics. Springer-Verlag. ISBN 978-0-387-98412-4 . Retrieved 2008-12-04. - "Die Homoiomorphie der kompakten konvexen Mengen im Hilbertschen Raum" [The homomorphism of the compact convex sets in Hilbert space] (in German). EUDML. Archived from the original on 2020-03-02.

- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978].
*Counterexamples in Topology*(Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.

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