Banach Space - Theorems and Properties

Theorems and Properties

• Theorem A normed space X is a Banach space if and only if each absolutely convergent series in X converges.

• Theorem Let X be a normed space. Then there is a Banach space Y and an isometric isomorphism T: X → Y such that T(X) is dense in Y. Furthermore, the space X′ is isometrically isomorphic to Y′. If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Z, then Z is isometrically isomorphic to Y.

• Theorem Let X and Y be normed spaces. Then, B(X, Y) := {T: X → Y | T linear and bounded}, is a normed space under the operator norm. If Y is a Banach space, then so is B(X, Y).

• Proposition Let T be a linear operator from a normed space X into a normed space Y. If X is a Banach space and T is an isomorphism, then T(X) is a Banach space.

• Corollary Every finite-dimensional normed space is a Banach space.

• Corollary A Banach space with a countable Hamel basis is finite-dimensional.

• The Open Mapping Theorem Let X and Y be Banach spaces and T: X → Y be a continuous linear operator. Then T is surjective if and only if T is an open map. In particular, if T is bijective and continuous, then T−1 is also continuous.

• Corollary Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.

• The Closed Graph Theorem Let T: X → Y be a linear function between Banach spaces. The graph is closed in X × Y if and only if T is continuous.

• Theorem If M is a closed subspace of a Banach space X, then X/M with is also a Banach space.

• The First Isomorphism theorem for Banach spaces Suppose that X and Y are Banach spaces and that T ∈ B(X, Y). Suppose further that the range of T is closed in Y. Then X/Ker(T) ≅ T(X). This is a topological isomorphism with means that a bijective, linear set L exists which goes from X/Ker(T) to T(X) so that both L and L−1 are continuous.
• Theorem Let X₁, ..., X_n be normed spaces. Then X₁ ⊕ ... ⊕ X_n is a Banach space if and only if each X_j is a Banach space.
• Proposition If X is a Banach space that is the internal direct sum of its closed subspaces M₁, ..., M_n, then X ≅ M₁ ⊕ ... ⊕ M_n.
• Theorem Every Banach space is a Fréchet space.
• Theorem For every separable Banach space X, there is a closed subspace M of ℓ₁ such that X ≅ ℓ₁/M.
• Hahn–Banach theorem Let X be a vector space of the field K. Let further
  • Y ⊆ X be a linear subspace,
  • p: X → R be a sublinear function and
  • f: Y → K be a linear functional so that Re f(y) ≤ p(y) for all y in Y.
    Then, there exists a linear functional F: X → K so that
  • and
  • .

In particular, every continuous linear functional on a subspace of a normed space can continuously be continued on the whole space.

• Banach–Steinhaus theorem Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. The uniform boundedness principle states that if for all x in X we have, then
• Banach–Alaoglu theorem Let X be a normed vector space. Then the closed unit ball of the dual space B′ := {x ∈ X′ | ǁxǁ ≤ 1} is compact in the weak* topology.