History, Developments and Open Problems on Approximation Properties

: In this paper, we give a comprehensive review of the classical approximation property. Then, we present some important results on modern variants, such as the weak bounded approximation property, the strong approximation property and p -approximation property. Most recent progress on E -approximation property and open problems are discussed at the end.

Proof. We take a separable reflexive Banach space X failing the approximation property and choose Z as in the theorem above for X * . Then Z * * * is isomorphic to Z * ⊕ X * * = Z * ⊕ X and fails the approximation property.
Grothendieck [6] systematically studied the variants of the approximation property. One important tool was the topology of uniform convergence on compact sets. Let τ be the topology of uniform convergence on compact sets on the space of bounded linear operators L(X, Y). It is generated by the seminorms T K = sup{ Tx : x ∈ K}, where K ranges over compact subsets of X. We write (L(X, Y), τ) for this space. Grothendieck characterized the dual space of (L(X, Y), τ).
Theorem 4 (Grothendieck). The continuous linear functionals on (L(X, Y), τ) consist of all functionals Φ of the form Grothendieck also gave a set of equivalent conditions for the approximation property.
Theorem 5 (Grothendieck). For a Banach space X, the following are equivalent: (1) X has the approximation property; (2) There is a net (T λ ) λ∈Λ of finite rank operators on X converging to the identity operator on X with respect to τ; (3) F (Y, X) is dense in (L(Y, X), τ), ∀Y; (4) F (X, Y) is dense in (L(X, Y), τ), ∀Y; (5) For every sequence of (x i ) ∈ X and (x * i ) ∈ X * such that ∑ x i x * i < ∞ and ∑ x * i (x)x i = 0, for all x ∈ X, we have ∑ x * i (x i ) = 0; (6) For every Banach space Y, every compact operator T ∈ L(Y, X) and > 0, there is a T 1 ∈ F (Y, X) such that T − T 1 < .
An immediate consequence of the theorem above is that if X does not have the approximation property, then we can find a separable subspace Y ⊂ X so that every space Z sitting between X and Z fails the approximation property. Another consequence is the following result. Theorem 6. X * has the approximation property implies that X has the approximation property. Therefore when X is reflexive space, X * has the approximation property if and only if X has the approximation property.
If we change the order of X and Y in (6), we get another theorem of Grothendieck.
Theorem 7. Let X be a Banach space. Then, X * has the approximation property if and only for every Banach space Y, every > 0 and every compact operator T ∈ L(X, Y), there is a finite rank operator T 1 : After Enflo constructed the first Banach space failing the approximation property, a number of important examples followed up. Figiel [14] and Davie [15,16] showed that there are subspaces of p , p > 2 which fail the approximation property and Szankowski [17] then found subspaces of p , 1 ≤ p < 2 failing the approximation property. All the proofs of these results used a general criterion of Enflo, which characterizes Banach spaces failing the approximation. Actually this criterion characterizes Banach spaces failing a weaker property-the compact approximation property. Definition 4. Let X be a Banach space. We say that X has the compact approximation property (CAP) if for every > 0 and for every compact set K ⊂ X, there exists a compact operator T on X so that Tx − x < for every x ∈ K, where K(X) is the space of compact operators on X.
Szankowski produced a fantastic set of examples of Banach spaces without the approximation property. The first example was the construction of a Banach lattice [18] failing the approximation property.
Szankowski's next construction [17] was of subspaces of p (p = 2) failing the approximation property. The importance of his construction is the explicit representation of the subspace in p . Theorem 9 (Szankowski). Let 1 ≤ p = 2. Then, there exists a subspace of p failing the compact approximation property.
Szankowski [19], using another spectacular construction, solved the following important problem. This space was very natural and is different from the previous constructed examples. Later, Godefroy and Saphar [20] showed that B(H)/K(H) also fails the approximation property. Theorem 11. The following conditions for a Banach space X are equivalent: (1) X has the bounded approximation property; (2) There is a net of finite rank operators (T α ) on X which tends strongly to the identity on X and satisfy sup α { T α } < ∞; (3) There exists a λ ≥ 1 so that for every finite dimensional subspace E of X, there is a finite rank operator T on X such that Tx = x and T ≤ λ, ∀x ∈ E.
For separable Banach spaces, we have the following corollary.
Corollary 2. Let X be a separable Banach space. Then, X has the λ-bounded approximation property if and only if there exists a sequence of operators (T n ) ⊂ F (X) converging to the identity in the strong operator topology so that lim sup n T n ≤ λ and T m T n = T n for all n < m.

Definition 5.
A Banach space X is said to have the λ-duality bounded approximation property if for every > 0 and every pair of finite dimensional subspaces F of X * and E of X, there exists an operator T ∈ F (X) with Tx − x ≤ x , for all x ∈ E, T ≤ λ, and T * x * − x * ≤ x * for all x * ∈ F.
In [21], Johnson proved the following result.
Theorem 12 (Johnson). If X * has the λ-bounded approximation property, then X has the λ-duality bounded approximation property.
The next theorem was given by Grothendieck [6].
Theorem 13 (Grothendieck). A separable dual space with the approximation property has the metric approximation property.
Using the theorem above, a result of Johnson [22] and a result of Lindenstrauss [23], we have the following corollary. Corollary 3. Every reflexive Banach space with the approximation property has the metric approximation property.
In [22], Johnson obtained a characterization of the bounded approximation property for non-separable spaces.
Theorem 14 (Johnson). A non-separable Banach space X has the λ-bounded approximation property if and only if every separable subspace of X is contained in a separable subspace with the λ-bounded approximation property.
The following is a result of Lindenstauss [23].
Theorem 15 (Lindenstrauss). Every reflexive Banach space has the separable complementation property with norm one projections. Question 3. If X is a (non-separable) dual space with the approximation property, does X have the metric approximation property? [24] proved the following: Theorem 16 (Figiel, Johnson). Let 1 ≤ λ < ∞. If X is a Banach space which has the λ-bounded approximation property in every equivalent norm, then X * has 2λ(1 + 4λ)-bounded approximation property.

Figiel and Johnson
In the case of the metric approximation property, the theorem above was strengthened by Johnson [22].
Theorem 17 (Johnson). If X has the metric approximation property in every equivalent norm, then X * has the metric approximation property. Corollary 4. Let X be a Banach space with a separable dual. Then the following are equivalent: (1) X * has the approximation property; (2) X * has the metric approximation property; (3) X has the metric approximation property in all equivalent norms.
With a little more efforts, we get another consequence of the theorems above.

Corollary 5.
(1) There exists a Banach space with the bounded approximation property but fails the metric approximation property. (2) There exists a separable Banach space with the approximation property but fails the bounded approximation property. Proof.
(1) Let X be a separable Banach space with a basis whose dual is separable and fails the approximation property. Then, for every n ∈ N, there is an equivalent norm | · | n on X so that (X, | · | n ) fails the n-bounded approximation property. Then, (X, | · | n ) has the bounded approximation property and fails the metric approximation property.
Here is another important open problem. Question 4. If a Banach space X has the bounded approximation property, does X have the metric approximation property in an equivalent norm?

Weak BAP and Strong AP
A long standing open problem in the study of approximation property is whether the BAP and the AP are equivalent for dual spaces.

Question 5. Are the BAP and AP equivalent for dual spaces?
In order to attack the problem, Oja and Lima introduced the strong AP and weak BAP which sit between AP and BAP. In 2008, Oja [3] introduced the strong approximation property. Definition 6. A Banach space X is said to have the strong approximation property (strong AP) if for every separable reflexive Banach space Y and every R ∈ K(X, Y), there exists a bounded net Lima and Oja [2] then introduced a weaker form of the BAP.

Definition 7.
For λ ≥ 1, a Banach space X is said to have the weak λ-bounded approximation property (weak λ-BAP) if for every Banach space Y and every weakly compact operator R : Theorem 18 (Oja). If X * has the AP, then X has the strong AP. If X has the strong AP, then X has the AP.
Theorem 19 (Lima, Oja). If X has the λ-BAP, then X has the weak λ-BAP. If X has the weak BAP, then X has the strong AP.
The formal implications between these approximation properties are: Theorem 20 (Lima and Oja [2]). The weak metric approximation property and the approximation property are equivalent for every dual space.
Theorem 21 (Oja [25]). The λ-bounded approximation property and the weak λ-bounded approximation property are equivalent for every Banach space with separable dual.
Lima and Oja [2] believed that the weak BAP and the BAP are not equivalent and Oja [3] conjectured that the strong AP and the weak BAP are not equivalent.
Oja's Conjecture. There exists a Banach space which has the strong AP but fails the weak BAP.
However, the authors [4] gave a negative answer to Oja's conjecture.
Theorem 22 (Kim and Zheng). The weak BAP and the strong AP are equivalent for every Banach space.
One of the main tools to prove the above theorem is the following: Theorem 23 (Kim and Zheng). Suppose that X * * or Y * has the Radon-Nikodym property. If X has the strong AP, then, for every R ∈ L(X, Y), there exists a λ R > 0 such that id X ∈ {S ∈ F (X) : The Radon-Nikodym property in the assumption is needed since we used a representation of the dual space of compact operators by Feder and Saphar [26].
Theorem 24 (Feder and Saphar). Suppose that X * * or Y * has the Radon-Nikodym property. If f ∈ (K(X, Y), · ) * , then for every ε > 0, there exist sequences (y * n ) in Y * and (x * * n ) in X * * with ∑ n y * n x * * n < f + ε such that f (U) = ∑ n x * * n (U * y * n ) for U ∈ K(X, Y). ). Let X be a Banach space. Then, the following statements are equivalent:

Lemma 1 (Kim and Zheng
(a) X has the strong AP; (b) For every separable reflexive Banach space Y and every R ∈ K(X, Y), there exists a λ R > 0 such that (c) For every Banach space Y and every R ∈ K(X, Y), there exists a λ R > 0 such that id X ∈ {S ∈ F (X) : Lemma 2. Let X be a Banach space and λ > 0. Then, X has the weak λ-BAP if and only if for every separable reflexive Banach space Y and every R ∈ K(X, Y),

Proof of the Main Theorem.
Suppose that X fails the weak BAP. Then, by the above lemma, for every m ∈ N, there exist a separable reflexive Banach space Y m and R m ∈ K(X, Y m ) such that We may assume R m = 1 for all m. Now, we define the map R : Then, the map is well defined and linear, and it is regular to check that R ∈ K(X, (∑ m ⊕Y m ) c 0 ). By the assumption and our previous lemma, there exists a λ R > 0 such that Hence for every m, we have which is a contradiction.
Recall that the formal implications between these approximation properties are: Figiel and Johnson [24] first showed that there is a Banach space with the AP but fails the BAP. Then Figiel, Johnson and Pelczynski [27] proved that c 0 and 1 have subspaces with the AP but failing the BAP. In 2015, this result was improved by Chen, Kim and Zheng [28].
Theorem 25. Each of c 0 and 1 has a subspace which has the AP but fails the weak BAP.
The proof of the above theorem uses the following result whose proof involves certain properties of the weak BAP and some of the techniques in [27].
Theorem 26 (Chen, Kim and Zheng). If X has the AP but X * fails the AP, then there exists a subspace Y of (∑ n X) c 0 (or (∑ n X) p (1 ≤ p < ∞)) such that Y has the AP but it fails to have the weak BAP. [27] defined the bounded approximation property of pairs. Definition 8. Let Y be a subspace of a Banach space X and λ ≥ 1. (X, Y) is said to have the λ-BAP if for every finite-dimensional subspace F of X and every ε > 0 there exists an S ∈ F (X) with S ≤ λ + ε such that Sx = x for all x ∈ F and S(Y) ⊂ Y.

Figiel, Johnson and Pelczyński
The pair (X, X) has the λ-BAP if and only if X has the λ-BAP [12]. This notion was first extended to the bounded compact approximation property for pairs by Chen and Zheng [29]. Then Chen, Kim and Zheng [28] defined a weaker notion of the bounded approximation property of pairs. Definition 9. The pair (X, Y) is said to have the weak λ-BAP if for every Banach space Z and R ∈ W (X, Z), for every finite-dimensional subspace F of X and for every ε > 0, there exists an S ∈ F (X) with RS ≤ (λ + ε) R such that Sx = x for all x ∈ F and S(Y) ⊂ Y. Theorem 27. Let X be a Banach space. Then the following statements are equivalent.
(a) id X * ∈ F * (X) τ c ; (b) For every finite-codimensional subspace Y of X, the pair (X, Y) has the weak λ-BAP; (c) There exists a λ ≥ 1 satisfying that for every finite-codimensional subspace W of X, there exists a finite-codimensional subspace Y of X with Y ⊂ W such that the pair (X, Y) has the weak λ-BAP.
Lemma 3. Let X be a Banach space. Let Y be a finite-codimensional subspace of X. If Y has the weak λ-BAP, then the pair (X, Y) has the weak 3λ-BAP.
Corollary 6. Let X be a Banach space. If id X * ∈ F * (X) τ c , then for any λ ≥ 1, there exists a finite-codimensional subspace Y λ of X such that Y λ does not have the weak λ-BAP.

Banach Approximation Property
Compactness is a topological property which also has geometric characterizations in Banach spaces. One of the most famous such characterizations is probably the Grothendieck's criterion [6], which can be stated as the following: A subset K of a Banach space X is relatively compact if and only if for every ε > 0, there exists where c 0 (X) is the space of sequences in X which converge to 0 in norm. We denote by B Z the closed unit ball of a Banach space Z.
Motivated by this, Sinha and Karn [30] introduced the notion of p-compact sets. Let 1 ≤ p < ∞ and let 1/p + 1/p * = 1. A subset K of X is said to be p-compact if there exists (x n ) n ∈ p (X), which is the Banach space with the norm · p of all X-valued absolutely p-summable sequences, such that K ⊂ p-co(x n ) n := ∞ ∑ n=1 α n x n : (α n ) ∈ B p * .
Every p-compact set is relatively compact. An operator T : Y → X is called p-compact if there exists (x n ) n ∈ p (X) such that T(B Y ) ⊂ p-co(x n ) n . The set of p-compact operators from Y into X is denoted by K p (Y, X) and the K p -norm is given by Delgado, Piñeiro and Serrano [31,32] showed that [K p , · K p ] is a Banach operator ideal. The concept of p-compact sets naturally leads to that of the p-AP.
Definition 10. For 1 ≤ p ≤ ∞, a Banach space X is said to have the p-AP if for every p-compact subset K of X and every > 0, there exists a finite rank operator T on X such that sup x∈K Tx − x ≤ .
In fact, the ∞-AP means the AP. We can see easily that if X has the q-AP, then X has the p-AP for 1 ≤ p < q ≤ ∞. An interesting result given by Sinha and Karn [30] is that every Banach space has the 2-AP and hence has the p-AP for every 1 ≤ p ≤ 2. In 2009, Delgado, Oja, Pineiro and Serrano [33] proved that if X * * has the p-AP, then X has the p-AP.
One year later, Choi and Kim [34] got the following nice result which is parallel to the result for AP. (Choi and Kim). Let X be a Banach space and 2 < p < ∞. If X * has the p-AP, then so does X.

Theorem 28
One of the crucial tools in their proof is a representation theorem of the dual of (L(X, Y), τ p ) which is an analogue of Grothendieck's representation theorem.
Theorem 29 (Choi and Kim). Let 1 < p < ∞. Then (L(X, Y), τ p ) * consists of all linear functionals f of the form In the same paper, Choi and Kim showed that the converse of the theorem is not true in general. They first used the Davie space [15] to prove the following: Theorem 30 (Choi and Kim). Let 2 < p < ∞. There exists a subspace q which fails the p-AP whenever q > 2p/(p − 2).
As a consequence of Theorems 30 and 3, Choi and Kim gave a new result, as we can see in the next theorem.
Theorem 31 (Choi and Kim). For every 2 < p < ∞, there exists a separable Banach space X p such that X * * p has a basis but X * * * p fails the p-AP.
Most recently, Kim and Zheng [35] introduced a more general type of approximation property called the E-approximation property and its variants. In the same paper, characterizations of these properties were given and representation theorems for the dual of L(X, Y) under the topology of uniform convergence on E-compact sets and E u -compact sets will also be presented.
Let E be a Banach space with a normalized basis (e n ) n and (e * n ) n denote the sequence of coordinate functionals. Let X be a Banach space and define E w (X) := (x n ) n in X : x * (x n )e n converges in E for each x * ∈ X * , x n e n converges in E .
When (e n ) n is 1-unconditional, K is said to be E-compact if there exists (x n ) n ∈ E(X) such that K ⊂ E-co(x n ) n . Definition 11. A Banach space X is said to have the E u -approximation property (E u -AP) (respectively, E-AP whenever (e n ) n is 1-unconditional) if for E u -compact subset (respectively, E-compact subset) K of X and every > 0, there exists an S ∈ F (X, X) such that It follows from the definition that the p-AP is simply the p -AP and AP ⇔ c 0 -AP ⇔ c u 0 -AP.
The following fact can be verified easily from the definition.

Lemma 4.
Every E u -compact set is a relatively compact set. If the basis (e n ) for E is 1-unconditional, then every E-compact set is E u -compact.
We denote by τ E u (respectively, τ E whenever (e n ) n is 1-unconditional) the locally convex topology on the space L(X, Y) of all operators from a Banach space X to a Banach space Y defined by the seminorms sup x∈K Tx , where the supremums are taken over all E u -compact (respectively, E-compact) subsets of X. Then, a Banach space X has the E u -AP (respectively, E-AP) is equivalent to When (e n ) n is 1-unconditional, since every E-compact set is an E u -compact set, τ E u is stronger than τ E .
An operator R : Y → X is said to be E u -compact (respectively, E-compact whenever (e n ) n is 1-unconditional) if R(B Y ) is an E u -compact (respectively, E-compact) subset of X. We denote by K E u (Y, X) (respectively, K E (Y, X)) the collection of all E u -compact (respectively, E-compact) operators from Y to X. Also, we let The following theorems characterize E u -AP and E-AP using E u -compact and E-compact operators [35].
Theorem 32 (Kim, Zheng). The following statements are equivalent for a Banach space X.
(a) X has the E u -AP; (b) For every Banach space Y and every R ∈ K E u (Y, X), (c) For every quotient space Y of E * and every injective R ∈ K E u (Y, X), Theorem 33 (Kim, Zheng). Suppose that (e n ) n is 1-unconditional. The following statements are equivalent for a Banach space X.
(a) X has the E-AP; (b) For every Banach space Y and every R ∈ K E (Y, X), R ∈ {SR : S ∈ F (X, X)} · K E u ; (c) For every quotient space Y of E * and every injective R ∈ K E (Y, X), Let X and Y be Banach spaces. We denote by X ⊗ Y the algebraic tensor product of X and Y. The normed space X ⊗ Y equipped with a norm α is denoted by X ⊗ α Y and its completion is denoted by X⊗ α Y. One of the most classical norms on X ⊗ Y is the injective tensor norm which is denoted by ε.
where ∑ m n=1 x n ⊗ y n is any representation of u. In [35], the following identification was attained.
Proposition 1 (Kim, Zheng). For any Banach space X, Recall that the projective tensor norm π is defined by for all u ∈ X ⊗ Y. X⊗ π Y is the completion of X ⊗ Y under the projective tensor norm. If u ∈ X⊗ π Y, then there exist sequences (x n ) n in X and (y n ) n in Y with ∑ ∞ n=1 x n y n < ∞ such that whenever X * is separable and has the AP. The next theorem is a characterization of the dual of L(X, Y) under the topology τ E u .
Theorem 34 (Kim, Zheng). Suppose that (e n ) n is shrinking. Let X and Y be Banach spaces. Then we have A similar result holds for the dual of L(X, Y) under the topology τ E .
Theorem 35 (Kim, Zheng). Suppose that (e n ) n is 1-unconditional and shrinking. Let X and Y be Banach spaces. Then we have Using the representations above, Kim and Zheng [35] obtained another characterization of the E u -AP when E has a shrinking basis.
Theorem 36 (Kim, Zheng). Suppose that (e n ) n is shrinking. The following statements are equivalent for a Banach space X.
(a) X has the E u -AP; (b) For every separable reflexive Banach space Y and every R ∈ K(X, Y), R ∈ {RS : S ∈ F (X, X)} τ E u ; (c) For every Banach space Y and every R ∈ K(X, Y), R ∈ {RS : S ∈ F (X, X)} τ E u ; (d) For every separable reflexive Banach space Y, A similar result was attained for E u -AP in [35].
Theorem 37 (Kim, Zheng). Suppose that (e n ) n is 1-unconditional and shrinking. The following statements are equivalent for a Banach space X.
An interesting application of the above theorem is the following result [35].
Theorem 38 (Kim, Zheng). Let 1 < p, q < ∞ and let X be a Banach space. If X * has the (∑ q ) p -AP, then X has the p-AP.

Remark 1.
In [34], one of the main theorems proved by Choi and Kim is that if X * has the p-AP, then X has the p-AP. In Theorem 38, if p = q, then we recover the theorem of Choi and Kim.
Theorem 39 (Kim, Zheng). Let 1 < p ≤ q < ∞ and let X be a Banach space. If X * has the (∑ p ) q -AP, then X has the p-AP.
Theorem 39 does not tell us what happens when q < p. So the problem below is left unsolved. Problem 1. Let 1 < q < p < ∞ and X be a Banach space. If X * has the (∑ p ) q -AP, does X have the p-AP?
Theorems 38 and 39 do not include the cases when p = 1, p = ∞, q = 1 or q = ∞. So the following problems are still open. Problem 2. Let 1 < q < ∞ and X be a Banach space. If X * has the (∑ q ) 1 -AP, does X have the q-AP?