Banach space

Banach spaces play a central role in functional analysis.

Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin.

is; an example[note 5] can even be found in a (non-complete) pre-Hilbert vector subspace of

is only a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is not associated with any particular norm or metric (both of which are "forgotten").

is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets.

Normable TVSs are characterized by being Hausdorff and having a bounded convex neighborhood of the origin.

into complete metrizable TVSes (for example, Banach or Fréchet spaces), if one topology is finer or coarser than the other, then they must be equal (that is, if

However, the topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of norms.

that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology

forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.

which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above).

The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.

[37][38] The result has been extended by Amir[39] and Cambern[40] to the case when the multiplicative Banach–Mazur distance between

In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual

is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood

This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps.

Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space

Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems.

basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.

The unit ball of the bidual is a pointwise compact subset of the first Baire class on

Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.

has a basis[58] remained open for more than forty years, until Bočkarev showed in 1974 that

obtained by extending the identity map of the algebraic tensor product.

Grothendieck related the approximation problem to the question of whether this map is one-to-one when

is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and

The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant

[70] The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies.

Kadec's theorem was extended by Torunczyk, who proved[76] that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.

Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:[77] Theorem[78] — Let

Glossary of symbols for the table below: Several concepts of a derivative may be defined on a Banach space.