1.
Schwartz space
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In mathematics, Schwartz space is the function space of functions all of whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space and this property enables one, by duality, to define the Fourier transform for elements in the dual space of S, that is, for tempered distributions. The Schwartz space was named in honour of Laurent Schwartz by Alexander Grothendieck, a function in the Schwartz space is sometimes called a Schwartz function. Here, sup denotes the supremum, and we again use multi-index notation, in particular, S is a subspace of the function space C∞ of infinitely differentiable functions. If i is a multi-index, and a is a real number. Any smooth function f with compact support is in S and this is clear since any derivative of f is continuous and supported in the support of f, so f has a maximum in Rn by the extreme value theorem. S is a Fréchet space over the complex numbers, from Leibniz rule, it follows that S is also closed under pointwise multiplication, if f, g ∈ S, then fg ∈ S. If 1 ≤ p ≤ ∞, then S ⊂ Lp, the space of all bump functions, C∞ c, is included in S. The Fourier transform is a linear isomorphism S → S, if f ∈ S, then f is uniformly continuous on R. Hörmander, L. The Analysis of Linear Partial Differential Operators I, methods of Modern Mathematical Physics, Functional Analysis I. This article incorporates material from Space of rapidly decreasing functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License

2.
Topological vector space
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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure with the concept of a vector space. Hilbert spaces and Banach spaces are well-known examples, unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. Some authors require the topology on X to be T1, it follows that the space is Hausdorff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. Every normed vector space has a topological structure, the norm induces a metric. This is a vector space because, The vector addition +, V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm, the scalar multiplication ·, K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the inequality and homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of vector spaces. There are topological spaces whose topology is not induced by a norm. These are all examples of Montel spaces, an infinite-dimensional Montel space is never normable. A topological field is a vector space over each of its subfields. A cartesian product of a family of vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f, R → R, with this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following, if is a sequence of elements in X, then fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable, indeed, every neighborhood of 0 in the topology contains lines