Functional analysis

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• ConceptTopological vector space

  A topological vector space is a vector space over $\R$ or $\C$ with a topology in which addition and scalar multiplication are continuous. It is the common frame of functional analysis: normed spaces, Banach spaces, Hilbert spaces and locally convex spaces are all examples. Under addition it is a commutative topological group, so its topology is determined by the neighbourhoods of $0$. Continuity of linear maps, convergence of series and duality are all studied in this setting.

• ConceptTopological group structure of a vector space

  Every topological vector space is, under addition, a commutative topological group. This additive structure is the source of many basic facts about topological vector spaces. Translations are homeomorphisms, so the whole topology is fixed by the neighbourhoods of $0$. It also gives each topological vector space a canonical uniform structure, so Cauchy nets and completeness make sense without a metric.

• ConceptSeminorm

  A seminorm on a real or complex vector space has all the properties of a norm, except that it may vanish on nonzero vectors. Seminorms are the building blocks of locally convex spaces: a family of seminorms defines a topology, and every locally convex topology arises in this way. Many function spaces are described by seminorms rather than by one norm. An example is the space of continuous functions on $\R$ with uniform convergence on compact sets.

• ConceptTopology generated by a family of seminorms

  A family of seminorms on a vector space defines a topology. In it, a net converges exactly when it converges for every seminorm of the family. With this topology the vector space is a topological vector space, and in fact a locally convex space. This is the standard way to put a topology on function spaces such as spaces of smooth functions, where no single norm is suitable.

• ConceptLocally convex space

  A locally convex space is a topological vector space in which every neighbourhood of $0$ contains a convex neighbourhood of $0$. Equivalently, its topology is generated by a family of seminorms. Normed spaces, Fréchet spaces, spaces of smooth functions and spaces of distributions are all locally convex. Local convexity is what makes the Hahn–Banach theorem work, so these spaces have many continuous linear forms and a rich duality theory.

• ConceptMinkowski functional

  The Minkowski functional, or gauge, of an absorbing set $K$ in a vector space is a function on the space. Its value at $x$ is the smallest factor by which $K$ must be scaled to contain $x$. For convex sets it is sublinear, and for convex balanced sets it is a seminorm. It is the bridge between convex neighbourhoods of $0$ and seminorms, and so it is the main tool for describing locally convex spaces by seminorms.