Auxiliary normed space

In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.[1] One method is used if the disk D is bounded: in this case, the auxiliary normed space is span D with norm p_D := r. The other method is used if the disk D is absorbing: in this case, the auxiliary normed space is the quotient space X / p^-1
_D(0). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces).

Preliminaries

A subset of a vector space is called a disk and is said to be disked, absolutely convex, or convex balanced if it is convex and balanced.

If C and D are subsets of a vector space X then D absorbs C if there exists a real r > 0 such that C ⊆ aD for any scalar a satisfying |a| ≥ r. WThe set D is called absorbing in X if D absorbs { x } for every x ∈ X.

A subset B of a topological vector space (TVS) X is said to be bounded in X if every neighborhood of the origin in X absorbs B. A subset of a TVS X is called bornivorous[2] if it absorbs all bounded subsets of X.

Induced by a bounded disk – Banach disks

Henceforth, X will be a real or complex vector space (not necessarily a TVS, yet) and D will be a disk in X.

Seminormed space induced by a disk

Let X will be a real or complex vector space. For any subset D of X, the Minkowski functional of D defined by:

• If D = ∅ then define p_∅ : { 0 } → [0, ∞) to be the trivial map p_∅ = 0[3] and it will be assumed that span ∅ := { 0}.[note 1]
• If D ≠ ∅ and if D is absorbing in span D then denote the Minkowski functional of D in span D by

  p_D : span D → [0, ∞)

  where for all x ∈ span D, this is defined by

  p_D(x) := inf { r : x ∈ rD, r > 0}.

Let X will be a real or complex vector space. For any subset D of X such that the Minkowski functional p_D is a seminorm on span D, let X_D denote

X_D := (span D, p_D)

which is called the seminormed space induced by D where it is say "normed" if p_D is a norm.

Assumption (Topology): X_D = span D is endowed with the seminorm topology induced by p_D, which will be denoted by 𝜏_D or 𝜏_pD

Importantly, this topology stems entirely from the set D, the algebraic structure of X, and the usual topology on ℝ (since p_D is defined using only the set D and scalar multiplication). This justifies the study of Banach disks and is part of the reason why they play an important role in the theory of nuclear operators and nuclear spaces.

The natural inclusion In_D : X_D → X is called the canonical map.[1]

Suppose that D is a disk. Then span D = ∪^∞
_n=1 nD so that D is absorbing in span D, the linear span of D. The set { rD : r > 0 } of all positive scalar multiples of D forms a basis of neighborhoods at 0 for a locally convex topological vector space topology 𝜏_D on span D. The Minkowski functional of the disk D in span D guarantees that p_D is well-defined and forms a seminorm on span D.[4] The locally convex topology topology induced by this seminorm is the topology 𝜏_D that was defined before.

Banach disk definition

A bounded disk D in a topological vector space X such that (X_D, p_D) is a Banach space is called a Banach disk, infracomplete, or a bounded completant in X.

If its shown that (span D, p_D) is a Banach space then D will be a Banach disk in any TVS that contains D as a bounded subset.

This is because the Minkowski functional p_D is defined in purely algebraic terms. Consequently, the question of whether or not (X_D, p_D) forms a Banach space is dependent only on the disk D and the Minkowski functional p_D, and not on any particular TVS topology that X may carry. Thus the requirement that a Banach disk in a TVS X be a bounded subset of X is the only property that ties a Banach disk's topology to the topology of its containing TVS X.

Properties of disk induced seminormed spaces

Bounded disks

The following result explains why Banach disks are required to be bounded.

Theorem[5][2][1] — If D is a disk in a topological vector space (TVS) X, then D is bounded in X if and only if the natural inclusion In_D : X_D → X is continuous.

Proof

If the disk D is bounded in the TVS X then for all neighborhoods U of 0 in X, there exists some r > 0 such that rD ⊆ U ∩ XD. It follows that in this case the topology of (XD, pD) is finer than the subspace topology that XD inherits from X, which implies that the natural inclusion InD : XD → X is continuous. Conversely, if X has a TVS topology such that InD : XD → X is continuous, then for every neighborhood U of 0 in X there exists some r > 0 such that rD ⊆ U ∩ XD, which shows that D is bounded in X.

Hausdorffness

The space (X_D, p_D) is Hausdorff if and only if p_D is a norm, which happens if and only if D does not contain any non-trivial vector subspace.[6] In particular, if there exists a Hausdorff TVS topology on X such that D is bounded in X then p_D is a norm. An example where X_D is not Hausdorff is obtained by letting X = ℝ² and letting D be the x-axis.

Convergence of nets

Suppose that D is a disk in X such that X_D is Hausdorff and let x_• = (x_i)_{i ∈ I} be a net in X_D. Then x_• → 0 in X_D if and only if there exists a net r_• = (r_i)_{i ∈ I} of real numbers such that r_• → 0 and x_i ∈ r_iD for all i; moreover, in this case it will be assumed without loss of generality that r_i ≥ 0 for all i.

Relationship between disk-induced spaces

If C ⊆ D ⊆ X then span C ⊆ span D and p_D ≤ p_C on span C, so define the following continuous[2] linear map:

If C and D are disks in X with C ⊆ D then call the natural inclusion In^D
_C : X_C → X_D the canonical inclusion of X_C into X_D.

In particular, the subspace topology that span C inherits from (X_D, p_D) is weaker than (X_C, p_C)'s seminorm topology.[2]

D as the closed unit ball

The disk D is a closed subset of (X_D, p_D) if and only if D is the closed unit ball of the seminorm p_D i.e. D = { x ∈ span D : p_D(x) ≤ 1 }.

If D is a disk in a vector space X and if there exists a TVS topology 𝜏 on span D such that D is a closed and bounded subset of (span D, 𝜏), then D is the closed unit ball of (X_D, p_D) (i.e. D = { x ∈ Span D : p_D(x) ≤ 1}) (see footnote for proof).[note 2]

Sufficient conditions for a Banach disk

The following theorem may be used to establish that (X_D, p_D) is a Banach space. Once this is established, D will be a Banach disk in any TVS in which D is bounded.

Theorem[7] — Let D be a disk in a vector space X. If there exists a Hausdorff TVS topology 𝜏 on span D such that D is a bounded sequentially complete subset of (span D, 𝜏), then (X_D, p_D) is a Banach space.

Proof

Assume without loss of generality that X = span D and let p := pD be the Minkowski functional of D. Since D is a bounded subset of a Hausdorff TVS, D do not contain any non-trivial vector subspace, which implies that p is a norm. Let 𝜏D denote the norm topology on X induced by p where since D is a bounded subset of (X, 𝜏), 𝜏D is finer than 𝜏. Because D is convex and balanced, for any 0 < m < n 2^-(n+1)D + ⋅⋅⋅ + 2^-(m+2)D = 2^-(m+1)(1-2^m-n) D ⊆ 2^-(m+2)D. Let x• = (xi)^∞ i=1 be a Cauchy sequence in (XD, p). By replacing x• with a subsequence, we may assume without loss of generality^† that for all i, xi+1 - xi ∈ 1/2^i+2 D. This implies that for any 0 < m < n, xn - xm = (xn - xn-1) + ⋅⋅⋅ + (xm+1 - xm) ∈ 2^-(n+1)D + ⋅⋅⋅ + 2^-(m+2)D ⊆ 2^-(m+2)D so that in particular, by taking m = 1 it follows that that x• is contained in x1 + 2^−3D. Since 𝜏D is finer than 𝜏, x• is a Cauchy sequence in (X, 𝜏). For all m > 0, 2^-(m+2)D is a Hausdorff sequentially complete subset of (X, 𝜏). In particular, this is true for x1 + 2^−3D so there exists some x ∈ x1 + 2^−3D such that x• → x in (X, 𝜏). Since xn - xm ∈ 2^-(m+2)D for all 0 < m < n, by fixing m and taking the limit (in (X, 𝜏)) as n → ∞, it follows that x - xm ∈ 2^-(m+2)D for each m > 0. This implies that p(x - xm) → 0 as m → ∞, which says exactly that x• → x in (XD, p). This shows that (XD, p) is complete. ^†This assumption is allowed because x• is a Cauchy sequence in a metric space (so the limits of all subsequences are equal) and a sequence in a metric space converges if and only if every subsequence has a sub-subsequence that converges.

Note that even if D is not a bounded and sequentially complete subset of any Hausdorff TVS, one might still be able to conclude that (X_D, p_D) is a Banach space by applying this theorem to some disk

K satisfying { x ∈ span D : p_D(x) < 1 } ⊆ K ⊆ { x ∈ span D : p_D(x) ≤ 1 }

since p_D = p_K.

The following are consequences of the above theorem:

• A sequentially complete bounded disk in a Hausdorff TVS is a Banach disk.[2]
• Any disk in a Hausdorff TVS that is complete and bounded (e.g. compact) is a Banach disk.[8]
• The closed unit ball in a Fréchet space is sequentially complete and thus a Banach disk.[2]

Suppose that D is a bounded disk in a TVS X.

• If L : X → Y is a continuous linear map and B ⊆ X is a Banach disk, then L(B) is a Banach disk and L|_XB : X_B → L(X_B) induces an isometric TVS-isomorphism Y_L(B) ≡ X_B / (X_B ∩ ker L).

Properties of Banach disks

Let X be a TVS and let D be a bounded disk in X.

If D is a bounded Banach disk in a Hausdorff locally convex space X and if T is a barrel in X then T absorbs D (i.e. there is a number r > 0 such that D ⊆ rT).[5]

If U is a convex balanced closed neighborhood of 0 in X then the collection of all neighborhoods rU, where r > 0 ranges over the positive real numbers, induces a topological vector space topology on X. When X has this topology, it is denoted by X_U. Since this topology is not necessarily Hausdorff nor complete, the completion of the Hausdorff space X / p^-1
_U(0) is denoted by X_U so that X_U is a complete Hausdorff space and p_U := r is a norm on this space making X_U into a Banach space. The polar of U, U°, is a weakly compact bounded equicontinuous disk in X' and so is infracomplete.

If X is a metrizable locally convex TVS then for every bounded subset B of X, there exists a bounded disk D in X such that B ⊆ X_D, and both X and X_D induce the same subspace topology on B.[2]

Induced by a radial disk – quotient

Suppose that X is a topological vector space and V is a convex balanced and radial set. Then { 1/n V : n = 1, 2, ...} is a neighborhood basis at the origin for some locally convex topology 𝜏_V on X. This TVS topology 𝜏_V is given by the Minkowski functional formed by V, p_V : X → ℝ, which is a seminorm on X defined by p_V := r. The topology 𝜏_V is Hausdorff if and only if p_V is a norm, or equivalently, if and only if X / p^-1
_V(0) = { 0 } or equivalently, for which it suffices that V be bounded in X. The topology 𝜏_V need not be Hausdorff but X / p^-1
_V(0) is Hausdorff. A norm on X / p^-1
_V(0) is given by ||x + p^-1
_V(0)|| := p_V(x), where this value is in fact independent of the representative of the equivalence class x + p^-1
_V(0) chosen. The normed space (X / p^-1
_V(0), ||⋅||)> is denoted by X_V and its completion is denoted by X_V.

If in addition V is bounded in X then the seminorm p_V : X → ℝ is a norm so in particular, p^-1
_V(0) = { 0}. In this case, we take X_V to be the vector space X instead of X / { 0 } so that the notation X_V is unambiguous (whether X_V denotes the space induced by a radial disk or the space induced by a bounded disk).[1]

The quotient topology 𝜏_Q on X / p^-1
_V(0) (inherited from X's original topology) is finer (in general, strictly finer) than the norm topology.

Canonical maps

The canonical map is the quotient map q_V : X → X_V = X / p^-1
_V(0), which is continuous when X_V has either the norm topology or the quotient topology.[1]

If U and V are radial disks such that U ⊆ V then p^-1
_U(0) ⊆ p^-1
_V(0) so there is a continuous linear surjective canonical map q_V,U : X / p^-1
_U(0) → X / p^-1
_V(0) = X_V defined by sending x + p^-1
_U(0) ∈ X_U = X / p^-1
_U(0) to the equivalence class x + p^-1
_V(0), where one may verify that the definition does not depend on the representative of the equivalence class x + p^-1
_U(0) that is chosen.[1] This canonical map has norm ≤ 1[1] and it has a unique continuous linear canonical extension to X_U that is denoted by q_V,U : X_U → X_V.

Suppose that in addition B ≠ ∅ and C are bounded disks in X with B ⊆ C so that X_B ⊆ X_C and the natural inclusion In^C
_B : X_B → X_C is a continuous linear map. Let In_B : X_B → X, In_C : X_C → X, and In^C
_B : X_B → X_C be the canonical maps. Then In_C = In^C
_B ∘ In_B and q_V = q_V,U ∘ q_U.[1]

Induced by a bounded radial disk

Suppose that S is a bounded radial disk. Since S is a bounded disk, if D := S then we may create the auxiliary normed space X_D = span D with norm p_D := r; since S is radial, X_S = X. Since S is a radial disk, if V := S then we may create the auxiliary seminormed space X / p^-1
_V(0) with the seminorm p_V := r; because S is bounded, this seminorm is a norm and p^-1
_V(0) = { 0 } so X / p^-1
_V(0) = X / { 0 } = X. Thus, in this case the two auxiliary normed spaces produced by these two different methods result in the same normed space.

Duality

Suppose that H is a weakly closed equicontinuous disk in X' (this implies that H is weakly compact) and let

U := H° = { x ∈ X : |h(x)| ≤ 1 for all h ∈ H}

be the polar of H. Since U° = H°° = H by the bipolar theorem, it follows that a continuous linear functional f belongs to X'_H = span H if and only if f belongs to the continuous dual space of (X, p_U), where p_U is the Minkowski functional of U defined by p_U := r.[9]

Related concepts

A disk in a TVS is called infrabornivorous[2] if it absorbs all Banach disks.

A linear map between two TVSs is called infrabounded[2] if it maps Banach disks to bounded disks.

Fast convergence

A sequence x_• = (x_i)^∞
_i=1 in a TVS X is said to be fast convergent[2] to a point x ∈ X if there exists a Banach disk D such that both x and the sequence is (eventually) contained in span D and x_• → x in (X_D, p_D).

Every fast convergent sequence is Mackey convergent.[2]

See also

• Bornological space – A topological vector space where any bounded linear operator into another space is always continuous
• Injective tensor product
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Nuclear operator
• Nuclear space – Type of topological vector space
• Initial topology – coarsest topology making certain functions continuous
• Projective tensor product
• Schwartz topological vector space
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
• Topological tensor product – Tensor product constructions for topological vector spaces
• Topological vector space – Vector space with a notion of nearness
• Ultrabornological space

Notes

1. This is the smallest vector space containing ∅. Alternatively, if D = ∅ then D may instead be replaced with { 0 }.
2. Assume WLOG that X = span D. Since D is closed in (X, 𝜏), it is also closed in (X_D, p_D) and since the seminorm p_D is the Minkowski functional of D, which is continuous on (X_D, p_D), it follows Narici & Beckenstein (2011, pp. 119–120) that D is the closed unit ball in (X_D, p).

References

1. Schaefer & Wolff 1999, p. 97.
2. Narici & Beckenstein 2011, pp. 441-457.
3. Schaefer & Wolff 1999, p. 169.
4. Trèves 2006, p. 370.
5. Trèves 2006, pp. 370-373.
6. Narici & Beckenstein 2011, pp. 115-154.
7. Narici & Beckenstein 2011, pp. 441-442.
8. Trèves 2006, pp. 370–371.
9. Trèves 2006, p. 477.

Bibliography

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External links

• Nuclear space at ncatlab