space consists of continuous functionals (or to ). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals. Banach spaces Banach spaces, introduced by Stefan Banach, are complete normed vector spaces. A first example is the vector space consisting of infinite vectors with real entries whose -norm given by The topologies on the infinite-dimensional space are inequivalent for different For example, the sequence of vectors in which the first components are and the following ones are converges to the zero vector for but does not for but More generally than sequences of real numbers, functions are endowed with a norm that replaces the above sum by the Lebesgue integral The space of integrable functions on a given domain (for example an interval) satisfying and equipped with this norm are called Lebesgue spaces, denoted These spaces are complete. (If one uses the Riemann integral instead, the space is complete, which may be seen as a justification for Lebesgue's integration theory.) Concretely this means that for any sequence of Lebesgue-integrable functions with satisfying the condition there exists a function belonging to the vector space such that Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces. Hilbert spaces Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert. The Hilbert space with inner product given by where denotes the complex conjugate of is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions by polynomials. By the Stone–Weierstrass theorem, every continuous function on can be approximated as closely as desired by a polynomial. A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space in the sense that the closure of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis of its cardinality is known as the Hilbert space dimension. Not only