In this paper we introduce two Bishop–Phelps–Bollobás type properties for bounded linear operators between two Banach spaces X and Y: property 1 and property 2. These properties are motivated by a Kim–Lee result which states, under our notation, that a Banach space X is uniformly convex if and only if the pair (X,K) satisfies property 2. Positive results of pairs of Banach spaces (X,Y) satisfying property 1 are given and concrete pairs of Banach spaces (X,Y) failing both properties are exhibited. A complete characterization of property 1 for the pairs (ℓp,ℓq) is also provided.

In this article, we study a version of the Bishop-Phelps-Bollobás property. We investigate a pair of Banach spaces (X, Y ) such that every operator from X into Y is approximated by operators which attains its norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop-Phelps-Bollobás point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs (X, Y ) which have and fail this property. Some stability results are obtained about 1 and ∞ sums of Banach spaces and we also study this property for bilinear mappings.2010 Mathematics Subject Classification. Primary: 46B04; Secondary: 46B20, 46B28, 46B25.

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In the first part of our paper, we show that ℓ ∞ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as ℓ 1 (c), also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of c 0 (ω 1 ) admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler in [8]) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a C ∞ -smooth norm. Finally, we prove that there is a dense subspace Y of ℓ c ∞ (ω 1 ) such that every renorming of Y contains an isometric copy of c 00 (ω 1 ).

We study approximation of operators between Banach spaces X and Y that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair (X, Y ) has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those X, called universal pointwise BPB domain spaces, such that (X, Y ) possesses pointwise BPB property for every Y , and on those Y , called universal pointwise BPB range spaces, such that (X, Y ) enjoys pointwise BPB property for every uniformly smooth X. We show that every universal pointwise BPB domain space is uniformly convex and that Lp(µ) spaces fail to have this property when p > 2. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators.arXiv:1709.00032v2 [math.FA] 26 Sep 2018 1 , Y ) fails the BPBp (even though all elements in L( 2 1 , Y ) attain their norms) [3]. Moreover, such Y can be found in a way that for every Banach space X, the set of all norm attaining operators is dense in L(X, Y ) [8, Theorem 4.2]. For more results and background on the BPBp, we refer the reader to the recent papers [1,6,9,10] and the references therein.In the very recent paper [13], the following stronger version of the Bishop-Phelps-Bollobás property was introduced (with the name of Bishop-Phelps-Bollobás point property).Definition 1.1 ([13, Definition 1.2]). We say that a pair (X, Y ) of Banach spaces has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property, for short) if given ε > 0, there existsη(ε) > 0 such that whenever T ∈ L(X, Y ) with T = 1 and x 0 ∈ S X satisfyThe difference between the BPBp and this new property is that, in the last one, the point x 0 does not move and the new operator S attains its norm at it. Also observe that, by an easy change of parameters, we may consider T ∈ L(X, Y ) with T 1 in the above definition, and we will use this fact without any explicit reference. Nevertheless, the same trick does not work for the point x 0 (as the operator S attains its norm at x 0 ), so we have to consider x 0 ∈ S X .Another useful definitions, motivated by the corresponding ones for the BPBp given in [8], are the following.Definition 1.2. A Banach space X is said to be a universal pointwise BPB domain space if (X, Y ) has the pointwise BPB property for every Banach space Y . A Banach space Y is said to be a universal pointwise BPB range space if (X, Y ) has the pointwise BPB property for every uniformly smooth Banach space X.As we already mentioned, the pointwise BPB property was introduced in [13] where, among other results, it was proved that if (X, Y ) has the pointwise BPB property then X must be uniformly smooth, that Hilbert spaces are universal pointwise BPB domain spaces, and that the class of universal pointwise BPB range spaces includes uni...