aa r X i v : . [ m a t h . G M ] S e p Transfinite dimensions
Gerald Kuba
Let | M | denote the cardinal number (the size ) of the set M , e.g. | N | = ℵ , | R | = 2 ℵ . Recall that | A | | B | is the cardinal number of the family of all functions from B to A . Note further thatif M is an infinite set then | M | is the size of the family of all finite subsets of M and | M | ℵ is the sizeof the family of all countable subsets of M while, of course, 2 | M | is the size of the family of all subsetsof M and | M | < | M | . Consequently, | M | ≤ | M | ℵ ≤ | M | . For a (real or complex) Hilbert space H let B be a linear basis of H and S be anorthonormal basis of H . Of course we have | B | ≥ | S | and | B | = | S | when B or S isfinite. Further it is well-known that | B | ≥ ℵ when S is infinite [4]. It is also known that | B | ℵ = | B | when B is infinite [6]. From the main theorem in [3] it follows that either | B | = | S | or | B | = 2 | S | . Unfortunately, this conclusion is not provable in standard settheory! Actually, the ”proof ” of the statement in [3] Theorem 1 uses a very strong cardinalhypothesis (namely the Generalized Continuum Hypothesis ) which is not mentioned in thestatement. Even worse, [3] Theorem 1 is false under the irrefutable assumption 2 ℵ > ℵ because then one immediately obtains the wrong conclusion | B | = | S | when | S | = ℵ .Fortunately, one need not assume an unprovable hypothesis in order to derive a naturaland simple relation between the linear dimension β = | B | and the orthonormal dimension σ = | S | of an arbitrary Hilbert space H . This relation between these two fundamentalconcepts of dimension , which can hardly be found in the literature, reads as follows. Theorem 1.
For every infinite-dimensional Hilbert space H we have β = |H| = σ ℵ .Remark. Concerning
Banach spaces as considered in [3] our Theorem 1 remains true when H is a (real or complex) Banach space with linear dimension β and a Schauder basis ofsize σ . In fact, this has been proved but unfortunately not stated in [3].As a consequence of Theorem 1, two non-isomorphic Hilbert spaces H and H can beisomorphic as pure vector spaces. For example let H = ℓ = ℓ ( N ) and H = ℓ ( R ) .(As usual, ℓ (Λ) consists of all mappings x : Λ → C such that x ( λ ) = 0 for onlycountably many λ ∈ Λ and P λ ∈ Λ | x ( λ ) | < ∞ .) Then we have β = β = 2 ℵ and σ = ℵ < ℵ = σ . Further examples are all pairs H = ℓ (Λ ) , H = ℓ (Λ )where | Λ | = κ , | Λ | = κ ℵ and κ = ℵ α + ω with an arbitrary ordinal α . (Then σ = κ < κ ℵ = σ = β = β .)If V is a vector space over a field K then let dim V denote its ordinary dimension . (Inparticular, dim H is the linear dimension β of the Hilbert space H .) For two vectorspaces V , V over one field we write V ֒ → V if and only if V contains an isomorphiccopy of V . For every field K and every set I the set K I of all functions from I to K becomes a vector space over K when the algebraic structure is defined in the canonical way.If I is a finite set then, of course, dim K I = | I | for every field K . But dim K I > | I | when I is infinite. Moreover, the dimension of the space K I can be determined precisely. Theorem 2. If V is an infinite vector space over an arbitrary field K such that K N ֒ → V or |V| > | K | then dim V = |V| . Corollary 1 [
The
Erd¨os - Kaplansky
Theorem ]. For every field K and every infinite set I , dim K I = | K | | I | . emark. A proof of the
Erd¨os - Kaplansky
Theorem is sketched in [2, Ch.II § Corollary 2.
For every field K and every infinite set I , dim K I = 2 | I | provided that | K | ≤ | I | . In particular, dim R N = 2 ℵ and dim R R = 2 ℵ .Remark. The inverse of Theorem 1 is false: For the polynomial ring V = Q [ X ] we havedim V = |V| = | Q | = ℵ and (hence by Corollary 1) Q N cannot be embedded in V .First we prove Theorem 2. In doing so the following proposition is essential. Proposition 1.
For every field K the set { ( a n ) n ∈ N | = a ∈ K } , which is obviously equipollent to K \ { } and hence equipollent to K when K is infinite,is a linearly independent subset of the vector space K N .Proof. Let a , a , ..., a m be distinct elements of K . We are done by verifying that forarbitrary λ , λ , ..., λ m ∈ K m X k =1 λ k · ( a nk ) n ∈ N is the zero vector (0 , , , ... ) in the space K N only in the trivial case λ = λ = · · · = λ m = 0 . In other words we have to verify that ∀ n ∈ N : m X k =1 λ k · a nk = 0implies λ = λ = · · · = λ m = 0 . Now, λ = λ = · · · = λ m = 0 is already a consequenceof the weaker assumption ∀ n ∈ { , , ..., m − } : m X k =1 λ k · a nk = 0because this system of equations has the matrix · · · · a a · · · · a m a a · · · · a m · · · · · · ·· · · · · · ·· · · · · · · a m − a m − · · · · a m − m and, naturally, the determinant of this matrix equals Y ≤ i
For A ⊂ I let A be the characteristic function of the set A . So A ( x ) ∈ { , } ⊂ K for all x ∈ I where A ( x ) = 1 when x ∈ A and A ( x ) = 0 when x A . Let F be a family of subsets of I with |F | = 2 | I | suchthat J I ∪ · · · ∪ I n whenever J ∈ F and I , ..., I n ∈ F \ { J } for arbitrary n ∈ N . (Such a family F exists by [5] Lemma 7.7.) Then n vectors in the space K I taken from the family G := { F | F ∈ F } must be linearly independent for arbitrary n ∈ N . This is true simply because the standard basis of the vector space K n consists of n linearly independent vectors. Hence dim K I ≥ |G| = |F | = 2 | I | . This is enough since | K I | ≤ (2 | I | ) | I | = 2 | I | . Remark.
Of course, K N ֒ → V if and only if dim V ≥ dim K N = | K | ℵ . But consider thespace V = K I where K is a field of size ℵ and | I | = ℵ . It is obvious that K N ֒ → V but it seems not possible to verify dim V ≥ | K | ℵ directly without using K N ֒ → V because the standard system of linearly independent vectors in V has size | I | = ℵ anda construction of ℵ linearly independent vectors (which is possible by using K N ֒ → V and Proposition 1) is not enough since it is undecidable whether ℵ ℵ is greater than orequal to ℵ . In view of the previous proof we have dim K I ≥ ℵ but for our purposethis is also not enough since 2 ℵ = ℵ ℵ and one cannot rule out ℵ ℵ < ℵ ℵ . Besides,the second assumption |V| > | K | of Theorem 2 is of no use in order to compute dim V because |V| > | K | means ℵ ℵ > ℵ which is unprovable.It remains to prove Theorem 1. To begin with we extend Theorem 2. Theorem 3. If V is a real or complex vector space such that the Hilbert space ℓ can bealgebraically embedded into V then dim V = |V| . In particular, dim ℓ = 2 ℵ . The proof of Theorem 3 is a simple adaption of the proof of Theorem 2 in view of the factthat the set { ( a n ) n ∈ N | = | a | < } is a subset of ℓ of size 2 ℵ which (by applyingProposition 1) is linearly independent. Remark.
Theorem 3 is useful to compute the dimension of several Banach spaces. Forexample, the dimension of the space of all bounded functions from an arbitrary infinite set X to R (or to C ) is equal to 2 | X | . Further it is a nice exercise to construct an embeddingof ℓ in order to show that the dimension of the space of all continuous functions from[0 ,
1] to R is equal to 2 ℵ . 3ith the help of Theorem 3 we immediately obtain Theorem 1. Let H be an infinite-dimensional Hilbert space. Then, naturally, ℓ ֒ → H and hence β = dim H = |H| . Itremains to verify |H| = σ ℵ where σ is the orthonormal dimension of H . Since H is norm-isomorphic to ℓ (Λ) with | Λ | = σ the proof of Theorem 1 is finished by showingthe following proposition. Proposition 2. | ℓ (Λ) | = | Λ | ℵ for every infinite index set Λ .Proof. Let K be either the field R or the field C . By definition, ℓ (Λ) = (cid:8) x ∈ A (Λ , K ) (cid:12)(cid:12) P λ ∈ Λ x ( λ ) < ∞ (cid:9) where A (Λ , K ) is the family of all functions x : Λ → K such that x ( λ ) = 0 for at mostcountably many λ ∈ Λ .Elementary transfinite arithmetics yields |A (Λ , K ) | = | Λ | ℵ | K | ℵ = | Λ | ℵ . Consequently, | ℓ (Λ) | ≤ | Λ | ℵ . It remains to verify | ℓ (Λ) | ≥ | Λ | ℵ .Let I (Λ) be the family of all injective functions from N to Λ . Naturally, |I (Λ) | = | Λ | ℵ .Obviously, the space ℓ (Λ) contains the set F (Λ) := (cid:8) x ∈ R Λ (cid:12)(cid:12) ∃ ϕ ∈ I (Λ) : x (Λ \ ϕ ( N )) = { } ∧ (cid:0) x ( ϕ ( n )) (cid:1) n ∈ N ∈ ∞ Q n =1 [0 , n ] (cid:9) .From |I (Λ) | = | Λ | ℵ and (cid:12)(cid:12) ∞ Q n =1 [0 , n ] (cid:12)(cid:12) = 2 ℵ we derive |F (Λ) | = | Λ | ℵ , q.e.d.Remark. Due to Anderson’s theorem [1], Hilbert space ℓ = ℓ ( N ) is homeomorphic tothe product space R N . And by Theorem 1 and Corollary 2 also the vector spaces ℓ and R N are isomorphic. However, Proposition 2 demonstrates that neither the first nor thesecond statement can be generalized from Λ = N to arbitrary index sets Λ . (If | Λ | = 2 κ for some transfinite cardinal κ then | ℓ (Λ) | = | Λ | ℵ = | Λ | < | Λ | = | R Λ | and hence ℓ (Λ)cannot be homeomorphic or algebraically isomorphic to R Λ .) References [1] Anderson, R.D.:
Hilbert space is homeomorphic to the countable infinite product oflines.
Bull. Amer. Math. Soc. , 515-519 (1966).[2] Bourbaki: Elements of Mathematics, Algebra I (2nd printing). Springer 1989.[3] Evans, J.W., and Tapia, R.A.: Hamel versus Schauder dimension.
Am. Math. Monthly , No.4, 385-388 (1970).[4] Halmos, P.R.: A Hilbert Space Problem Book. Springer 1974.[5] Jech, Th.: Set Theory. Third Millennium Edition. Springer 2002.[6] Kruse, A.H.: Badly incomplete normed linear spaces.
Math. Z. , 314-320 (1964). Author’s address:
Institute of Mathematics.University of Natural Resources and Life Sciences, Vienna, Austria.
E-mail: gerald.kuba(at)boku.ac.atgerald.kuba(at)boku.ac.at