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3 Sequence Spaces In this section we will study the sequence spaces with the norms in case of 1 ≤ p < ∞ and l p := {x ∈ F N : ||x|| p < ∞} √ √√√ ∑ ∞ ||x|| p := p |x i | p i=0 ||x|| ∞ := sup |x i | i∈N in case of p = ∞ for all x = (x i ) i∈N , as they are known in functional analysis. One important duality property of these spaces is expressed by the following theorem (see, for instance, Theorem II.2.3 in [9]): Theorem 2 (Landau). Let p,q > 1 be real numbers such that 1 p + 1 q = 1 or p = 1 and q = ∞. Then the map λ : l q → l ′ p,a ↦→ λ a with λ a : l p → F, (x k ) k∈N ↦→ ∑ ∞ k=0 a kx k is an isometric isomorphism. The map λ is also isometric in case of p = ∞ and q = 1. The proof is mainly based on Hölder’s Inequality. It is known that the fact that λ is an isomorphism cannot be generalized to the case p = ∞ and q = 1 straightforwardly, since the result depends on the underlying axiomatic setting in this case. On the one hand, using the Hahn-Banach Theorem one can extend the limit functional lim : c → F on the space of convergent sequences c to a functional L : l ∞ → F with the same norm and it is easy to see that this functional cannot be represented as λ a with some a ∈ l 1 . Thus we obtain the following classical property of the map λ defined in Landau’s Theorem (see, for instance, Theorem II.1.11 in [9]): Theorem 3. In ZF+AC the map λ : l 1 → l ′ ∞ is not surjective. Thus, one could say that l ′ ∞ is a proper superset of l 1 . On the other hand, Pincus proved a result, first stated by Solovay, which shows that the situation changes if we replace the Axiom of Choice by Dependent Choice and the Baire Property (see 29.37 in [5]): Theorem 4 (Solovay, Pincus). In ZF+DC+BP the map λ : l 1 → l ′ ∞ is an isometric isomorphism. The Theorem of Landau and its counterpart for the case p = ∞ and q = 1 have certain consequences concerning the existence of compatible representations of the sequence spaces. As a preparation we prove a characterization of weak convergence for these spaces. We recall that in functional analysis weak convergence means convergence with respect to the weak topology, i.e. a sequence (x n ) n∈N in a normed space X is said to converge weakly to x, if (f(x n )) n∈N converges to
f(x) for any linear bounded functional f : X → F. The first part of the following lemma is a known fact. We include the proof in order to indicate how the second part follows from the previous theorem. Lemma 2. Let 1 < p < ∞. A sequence ((x ij ) j∈N ) i∈N in l p converges weakly to (x j ) j∈N , if and only if the sequence converges with respect to the product topology on F N to (x j ) j∈N and if it is bounded in || || p . For p = ∞, the equivalence holds in ZF+DC+BP, whereas in ZF+AC merely the only-if-part is true. Proof. Let ((x ij ) j∈N ) i∈N be a sequence in l p which converges weakly to (x j ) j∈N ∈ l p , i.e. (f((x ij ) j∈N ) i∈N converges for any linear bounded functional f : l p → F to f((x j ) j∈N ). Since the canonical projections pr j : l p → F,(y j ) j∈N → y j are linear bounded functionals, it follows that (pr j ((x ij ) j∈N )) i∈N = (x ij ) i∈N converges for any fixed j ∈ N to pr j ((x j ) j∈N ) = x j and hence ((x ij ) j∈N ) i∈N converges with respect to the product topology on F N to (x j ) j∈N . Moreover, it is known that any weakly convergent sequence in l p is bounded. (This is a consequence of the Uniform Boundedness Theorem, see for instance Korollar IV.2.3 in [9], and can be proven in ZF+DC.) Now let us assume that ((x ij ) j∈N ) i∈N is a sequence in l p which converges to (x j ) j∈N ∈ l p with respect to the product topology on F N and which is bounded in || || p . We have to prove that the sequence (f((x ij ) j∈N )) i∈N converges for any functional f : l p → F to f((x j ) j∈N ). Let q be such that 1/p+1/q = 1 or q = 1 in case of p = ∞. If the map λ : l q → l ′ p from Landau’s Theorem 2 is an isometric isomorphism, then it suffices to prove that (λ a ((x ij ) j∈N )) i∈N = ( ∑ ∞ j=0 a jx ij ) i∈N converges for any a = (a j ) j∈N ∈ l q to λ a ((x j ) j∈N ) = ∑ ∞ j=0 a jx j . Therefore, let a = (a j ) j∈N ∈ l q , i.e. ||a|| q = ( ∑ ∞ j=0 |a j| q ) 1/q < ∞. Since ((x ij ) j∈N ) i∈N is bounded in l p , it follows that S := sup i∈N ||(x ij ) j∈N − (x j ) j∈N || p + 1 exists. Let ε > 0. There is some J ∈ N such that ( ∑ ∞ j=J+1 |a j| q ) 1/q < ε/(2S). Let M := max{|a 0 |, |a 1 |,..., |a J |}. Since ((x ij ) j∈N ) i∈N converges to (x j ) j∈N with respect to the product topology on F N there is some I ∈ N such that |x ij − x j | < ε/(2M(J+1)) for all i ≥ I and j = 0,...,J. Now we obtain by Hölder’s Inequality for all i ≥ I ∣ ≤ ≤ ∞∑ a j x ij − ∣ j=0 ∣ ∣∣∣∣∣ ∞∑ a j x j j=0 J∑ |a j · (x ij − x j )| + j=0 ∞∑ j=J+1 |a j · (x ij − x j )| J∑ |a j | · |x ij − x j | + ||(0,...,0,a J+1 ,a J+2 ,...)|| q · ||(x ij − x j ) j∈N || p j=0 ≤ (J + 1)M · ε 2M(J + 1) + ε 2S · S = ε.
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Page 13 and 14: For the compact-open topology on X
Page 15: 5. Schechter, E.: Handbook of Analy

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