In mathematics, specifically functional analysis, a Banach space is said to have the **approximation property (AP)**, if every compact operator is a limit of finite-rank operators. The converse is always true.

Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955).

Later many other counterexamples were found. The space of bounded operators on does not have the approximation property (Szankowski). The spaces for and (see Sequence space) have closed subspaces that do not have the approximation property.

A locally convex topological vector space *X* is said to have **the approximation property**, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank.^{ [2] }

For a locally convex space *X*, the following are equivalent:^{ [2] }

*X*has the approximation property;- the closure of in contains the identity map ;
- is dense in ;
- for every locally convex space
*Y*, is dense in ; - for every locally convex space
*Y*, is dense in ;

where denotes the space of continuous linear operators from *X* to *Y* endowed with the topology of uniform convergence on pre-compact subsets of *X*.

If *X* is a Banach space this requirement becomes that for every compact set and every , there is an operator of finite rank so that , for every .

Some other flavours of the AP are studied:

Let be a Banach space and let . We say that *X* has the *-approximation property* (**-AP**), if, for every compact set and every , there is an operator of finite rank so that , for every , and .

A Banach space is said to have **bounded approximation property** (**BAP**), if it has the -AP for some .

A Banach space is said to have **metric approximation property** (**MAP**), if it is 1-AP.

A Banach space is said to have **compact approximation property** (**CAP**), if in the definition of AP an operator of finite rank is replaced with a compact operator.

- Every subspace of an arbitrary product of Hilbert spaces possesses the approximation property.
^{ [2] }In particular,- every Hilbert space has the approximation property.
- every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, possesses the approximation property.
^{ [2] } - every nuclear space possesses the approximation property.

- Every separable Frechet space that contains a Schauder basis possesses the approximation property.
^{ [2] } - Every space with a Schauder basis has the AP (we can use the projections associated to the base as the 's in the definition), thus many spaces with the AP can be found. For example, the spaces, or the symmetric Tsirelson space.

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, specifically in functional analysis, a **C ^{∗}-algebra** is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra

In mathematics, the ** L^{p} spaces** are function spaces defined using a natural generalization of the

In mathematics, a **trace-class** operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces.

In the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from *X* into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space *X* is reflexive if and only if the canonical evaluation map from *X* into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a *non*-reflexive Banach space that is isometrically isomorphic to its bidual.

In functional analysis, a branch of mathematics, a **compact operator** is a linear operator *L* from a Banach space *X* to another Banach space *Y*, such that the image under *L* of any bounded subset of *X* is a relatively compact subset of *Y*. Such an operator is necessarily a bounded operator, and so continuous.

In mathematics, especially in functional analysis, the **Tsirelson space** is the first example of a Banach space in which neither an ℓ^{ p} space nor a *c*_{0} space can be embedded. The Tsirelson space is reflexive.

In the field of mathematics known as functional analysis, the **invariant subspace problem** is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces.

In mathematics, a **Hilbert–Schmidt operator**, named for David Hilbert and Erhard Schmidt, is a bounded operator *A* on a Hilbert space *H* with finite **Hilbert–Schmidt norm**

In mathematics, there are usually many different ways to construct a **topological tensor product** of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.

In mathematics, a **Schauder basis** or **countable basis** is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces.

In functional analysis, the concept of a **compact operator on Hilbert space** is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

**Per H. Enflo** is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental. Three of these problems had been open for more than forty years:

In mathematics, a **symmetrizable compact operator** is a compact operator on a Hilbert space that can be composed with a positive operator with trivial kernel to produce a self-adjoint operator. Such operators arose naturally in the work on integral operators of Hilbert, Korn, Lichtenstein and Marty required to solve elliptic boundary value problems on bounded domains in Euclidean space. Between the late 1940s and early 1960s the techniques, previously developed as part of classical potential theory, were abstracted within operator theory by various mathematicians, including M. G. Krein, William T. Reid, Peter Lax and Jean Dieudonné. Fredholm theory already implies that any element of the spectrum is an eigenvalue. The main results assert that the spectral theory of these operators is similar to that of compact self-adjoint operators: any spectral value is real; they form a sequence tending to zero; any generalized eigenvector is an eigenvector; and the eigenvectors span a dense subspace of the Hilbert space.

The strongest locally convex topological vector space (TVS) topology on , the tensor product of two locally convex TVSs, making the canonical map continuous is called the **projective topology** or the **π-topology**. When *X ⊗ Y* is endowed with this topology then it is denoted by and called the **projective tensor product** of *X* and *Y*.

In mathematics, the **injective tensor product** of two topological vector spaces was introduced by Alexander Grothendieck and was used by him to define nuclear spaces.

Nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

An **integral bilinear form** is a bilinear functional that belongs to the continuous dual space of , the injective tensor product of the locally convex topological vector spaces (TVSs) *X* and *Y*. An **integral linear operator** is a continuous linear operator that arises in a canonical way from an integral bilinear form.

The strongest locally convex topological vector space (TVS) topology on , the tensor product of two locally convex TVSs, making the canonical map *separately* continuous is called the **inductive topology** or the **ι-topology**. When *X ⊗ Y* is endowed with this topology then it is denoted by and called the **inductive tensor product** of *X* and *Y*.

This is a glossary for the terminology in a mathematical field of functional analysis.

- ↑ Megginson, Robert E.
*An Introduction to Banach Space Theory*p. 336 - 1 2 3 4 5 Schaefer & Wolff 1999, p. 108-115.

- Bartle, R. G. (1977). "MR0402468 (53 #6288) (Review of Per Enflo's "A counterexample to the approximation problem in Banach spaces"
*Acta Mathematica*130 (1973), 309–317)".*Mathematical Reviews*. MR 0402468. - Enflo, P.: A counterexample to the approximation property in Banach spaces.
*Acta Math.*130, 309–317(1973). - Grothendieck, A.:
*Produits tensoriels topologiques et espaces nucleaires*. Memo. Amer. Math. Soc. 16 (1955). - Halmos, Paul R. (1978). "Schauder bases".
*American Mathematical Monthly*.**85**(4): 256–257. doi:10.2307/2321165. JSTOR 2321165. MR 0488901. - Paul R. Halmos, "Has progress in mathematics slowed down?"
*Amer. Math. Monthly*97 (1990), no. 7, 561—588. MR 1066321 - William B. Johnson "Complementably universal separable Banach spaces" in Robert G. Bartle (ed.), 1980
*Studies in functional analysis*, Mathematical Association of America. - Kwapień, S. "On Enflo's example of a Banach space without the approximation property". Séminaire Goulaouic–Schwartz 1972—1973: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, 9 pp. Centre de Math., École Polytech., Paris, 1973. MR 407569
- Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977.
- Nedevski, P.; Trojanski, S. (1973). "P. Enflo solved in the negative Banach's problem on the existence of a basis for every separable Banach space".
*Fiz.-Mat. Spis. Bulgar. Akad. Nauk*.**16**(49): 134–138. MR 0458132. - Pietsch, Albrecht (2007).
*History of Banach spaces and linear operators*. Boston, MA: Birkhäuser Boston, Inc. pp. xxiv+855 pp. ISBN 978-0-8176-4367-6. MR 2300779. - Karen Saxe,
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*Topological Vector Spaces*. GTM.**3**. New York: Springer-Verlag. ISBN 9780387987262. - Singer, Ivan.
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