document: let @xmath0 and @xmath1 be banach spaces over a real or complex field @xmath6 . 
   we use the traditional notations @xmath7 and @xmath8 for the unit sphere and the closed unit ball of the space @xmath0 , respectively . 
   the banach space of all bounded linear operators @xmath9 will be represented by @xmath10 . 
   in particular , when @xmath11 we denote @xmath12 simply by putting @xmath13 called the dual space of @xmath0 . 
   we say that an operator @xmath14 attains its norm if there exists @xmath15 such that @xmath16 . in this case 
   , we say that @xmath17 is a norm attaining operator and it attains its norm at @xmath18 . 
   the subset of @xmath10 of all norm attaining operators is denoted by @xmath19 . 
   we recall that a bounded linear operator is compact if the closure of the image of the unit ball is compact . for @xmath20 
   we denote by @xmath21 the euclidean space @xmath22 endowed with the @xmath23-norm @xmath24 with @xmath25 and @xmath26 and @xmath27 endowed with the sup - norm @xmath28 . to simplify the notation we put just @xmath29 when the field that we are working is specified . 
   the bishop - phelps theorem @xcite says that every bounded linear functional can be approximated by norm attaining functionals . 
   in other words , this means that the set of all norm attaining functionals on a banach space @xmath0 is dense in its dual space @xmath13 . 
   it was proved by lindenstrauss @xcite in 1963 that , in general , the same result does not work for bounded linear operators . 
   more precisely , he exhibited a banach space @xmath0 such that the set @xmath30 is not dense in @xmath31 . 
   seven years later , bollobs proved a numerical version of the bishop - phelps theorem which nowdays is known as the bishop - phelps - bollobs theorem @xcite . as a consequence of ( 
   * theorem 2.1 ) we may enunciate this theorem as follows .    _ ( bishop - phelps - bollobs theorem , @xcite , @xcite ) _ let @xmath32 and suppose that @xmath33 and @xmath34 satisfy @xmath35 then , there are @xmath36 and @xmath37 such that @xmath38    since we do not have a bishop - phelps version for bounded linear operators , we can not expect a bishop - phelps - bollobs version for this type of functions either . 
   so it is natural to study the conditions that the banach spaces @xmath0 and @xmath1 must satisfy to get a theorem of this nature . in 2008 , acosta , aron , garca and maestre introduced a definition in order to attain this problem . 
   [ bpbp ] ( bishop - phelps - bollobs property , @xcite ) we say that a pair of banach spaces @xmath4 satisfies the _ bishop - phelps - bollobs property _ 
   ( bpbp , for short ) when given @xmath39 , there exists @xmath40 such that whenever @xmath41 and @xmath15 are such that @xmath42 there are @xmath43 and @xmath36 such that @xmath44    there are many classical banach spaces that satisfy the bpbp . 
   for example , when @xmath0 and @xmath1 are finite dimensional spaces , the pair @xmath4 has this property ( * ? ? ? * proposition 2.4 ) . 
   also , if @xmath1 has the property @xmath45 of lindenstrauss , as @xmath46 and @xmath47 do , then the pair @xmath4 satisfies the bpbp for all banach space @xmath0 ( * ? ? ? 
   * theorem 2.2 ) . 
   more positive results appear when we assume that the range space @xmath1 is uniformly convex : the pairs @xmath48 , @xmath49 and @xmath50 all satisfy the bpbp ( see @xcite , @xcite and @xcite , respectively ) . 
   also if @xmath0 is a uniformly convex banach space , then the pair @xmath4 has the bpbp for all banach space @xmath1 ( * ? ? ? * theorem 3.1 ) . 
   just to help make the paper entirely accessible , we remember the concept of uniform convexity . 
   a banach space @xmath0 is uniformly convex if given @xmath39 , there exists @xmath51 such that whenever @xmath52 satisfy @xmath53 , then @xmath54 
   . we recall that if @xmath55 then @xmath56 is uniformly convex .    in 2014 , 
   kim and lee ( * ? ? ? 
   * theorem 2.1 ) given a characterization for uniformly convex banach spaces that associate this type of spaces with a peculiarity on the bishop - phelps - bollobs property . 
   more precisely , they proved that a banach space @xmath0 is uniformly convex if and only if given @xmath39 then we are able to find a positive number @xmath40 such that whenever @xmath57 and @xmath15 satisfy the relation @xmath58 there exists a vector @xmath36 such that @xmath59 note that the theorem says that a banach space @xmath0 is uniformly convex if and only if the pair @xmath60 satisfies the bishop - phelps - bollobs property without changing the initial functional @xmath61 , that is , the functional that almost attains its norm at some point @xmath18 is the same that attains its norm at the new vector that is close to @xmath18 .    in this paper , we study this last result for bounded linear operators and we call it as the strong bishop - phels - bollobs property ( sbpbp ) . 
   first , we study it in the case that the real number @xmath62 depends of @xmath39 and also of a fixed operator @xmath17 , and we get some positives results about it . after that , we study the property in the uniform case , that is , when the number @xmath63 depends only of @xmath39 as we are used to work when we work with the bpbp . as we will see in the next section , we get many negatives results about the uniform case and we use these results to prove that there are uniformly convex banach spaces @xmath0 and infinite dimensional banach spaces @xmath1 such that the pair @xmath4 fails the sbpbp . finally , we give a complete characterization to the sbpbp for the pair @xmath5 which describes when these pairs satisfy the property . 
   in this section we study the strong bishop - phelps - bollobs property . 
   namely , we study the conditions that the banach spaces @xmath0 and @xmath1 must have and the hypothesis that we have to add to get a kim - lee type theorem for bounded linear operators . the kim - lee theorem is enunciated as follows . 
   [ kim - lee ] _ 
   ( kim - lee theorem , @xcite ) _ a banach space @xmath0 is uniformly convex if and only if given @xmath39 , there exists @xmath40 such that whenever @xmath57 and @xmath15 satisfy @xmath58 there is @xmath36 such that @xmath64    as we mentioned before , it is like a bishop - phelps - bollobs property without changing the initial functional @xmath61 . 
   a natural question arises : the result is true for other pairs of banach spaces @xmath4 with @xmath0 and @xmath1 having additional hypothesis considering bounded linear operators instead bounded linear functionals ? 
   although it is more natural put the question just like that , it is seems to us to be a strong problem in the sense that it will be hard to find concrete pairs of banach spaces @xmath4 satisfying the kim - lee theorem for bounded linear operators . 
   so we start by considering that the positive real number @xmath65 that appears in theorem [ kim - lee ] does not depend only of @xmath39 but also of a given operator @xmath17 fixed . 
   before we do that , we want to comment that carando , lassalle and mazzitelli @xcite defined a weak bpbp for ideals of multilinear mappings where the real positive number @xmath66 in the definition of the bpbp in this context depends of a given @xmath39 and also of the ideal norm of the operator defined on a normed ideal of @xmath67-linear mappings . in other words , a normed ideal of @xmath67-linear mappings @xmath68 where @xmath69 are banach spaces 
   has the weak bpbp if for each @xmath70 with @xmath71 and @xmath39 , there exists @xmath72 depending also of @xmath73 such that if @xmath74 satisfies @xmath75 , then there exist @xmath76 with @xmath77 and @xmath78 such that @xmath79 , @xmath80 and @xmath81 . 
   they proved that if @xmath82 are uniformly convex banach spaces then @xmath83 has the weak bpbp for ideals of multilinear mappings for all banach space @xmath1 . here 
   we will work on a different context where @xmath62 depends of a fixed operator not of the norm of the operators ideal as we may see in the following definition . 
   [ sbpbp ] a pair of banach spaces @xmath4 has the _ strong bishop - phelps - bollobs property _ ( or sbpbp for short ) if given @xmath39 and @xmath41 , there exists @xmath84 such that whenever @xmath15 satisfies @xmath85 there is @xmath36 such that @xmath86 if @xmath14 is compact and the pair @xmath4 has the sbpbp , then we say that the pair @xmath4 has the sbpbp for compact operators .    in the next proposition , we assume that the domain space @xmath0 is finite dimensional to get the sbpbp first result . 
   in fact , as we mentianed in the previous paragraph , we get a positive real number @xmath84 that depends of @xmath39 and also of the operator @xmath41 fixed at the beggining of the proof . 
   the proof is by contradiction . 
   let @xmath41 . 
   if the result is false for some @xmath87 , then for all @xmath88 , there exists @xmath89 such that @xmath90 but @xmath91 , where @xmath92 for @xmath88 . 
   since @xmath0 is finite dimensional , there exists a subsequence @xmath93 of @xmath94 such that @xmath95 for some @xmath96 . 
   this implies that @xmath97 and since @xmath98 we get that @xmath99 and so @xmath100 . 
   then @xmath101 and @xmath102 which is a contradiction . 
   the proof is again by contradiction . 
   let @xmath41 be a compact operator . 
   if the result is false , for some @xmath87 and for all @xmath88 , there exists @xmath89 such that @xmath104 but @xmath91 for all @xmath88 . 
   then @xmath105 as @xmath106 
   . since @xmath0 is uniformly convex , @xmath0 reflexive and then by the smulian theorem there exists a subsequence @xmath93 of @xmath94 and @xmath96 such that @xmath93 converges weakly to @xmath18 . since @xmath17 is completely continuous , @xmath107 converges in norm to @xmath108 as @xmath109 . 
   therefore @xmath110 and so @xmath100 . 
   thus @xmath101 and @xmath111 this implies that @xmath112 and , using again that @xmath0 is uniformly convex , we get that @xmath113 which is a contradiction because of the following inequalities : @xmath114    consequently we get two more positives results about the sbpbp . in the next corollary 
   we prove that the pair @xmath4 has the sbpbp whenever @xmath0 is uniformly convex and @xmath1 has the schur s property which implies as a particular case that the pair @xmath115 satisfies the property . 
   corollary [ sbpbp3 ] shows that whenever @xmath0 is uniformly convex and @xmath1 has finite dimension , the pair @xmath4 has the sbpbp by using the fact that every bounded linear operator with finite dimensional range is compact . 
   we apply theorem [ sbpbp2 ] . to do this , we prove that every bounded linear operator @xmath103 is compact . 
   indeed , since @xmath17 is continuous , @xmath17 is @xmath116-@xmath116 continuous . 
   let @xmath117 . 
   since @xmath0 is reflexive , by the smulian theorem , there are a subsequence of @xmath118 ( which we denote again by @xmath118 ) and @xmath96 such that @xmath119 . 
   so @xmath120 . 
   now , since @xmath1 has the schur s property , @xmath121 in norm . 
   so @xmath17 is compact . by theorem [ sbpbp2 ] 
   the pair @xmath4 has the sbpbp . 
   [ sbpbp5 ] note that in definition [ sbpbp ] the operator @xmath103 must attains its norm if the pair @xmath4 has the sbpbp . 
   so if @xmath0 is not reflexive , then the pair @xmath4 fails the sbpbp for all banach space @xmath1 . 
   indeed , since @xmath0 is not reflexive , by the james theorem , there is a linear continuous functional @xmath122 such that @xmath123 for all @xmath124 . 
   let @xmath125 and define @xmath103 by @xmath126 
   . then @xmath127 and @xmath128 for all @xmath124 . 
   this implies that @xmath17 never attains its norm and then the pair @xmath4 can not have the sbpbp . 
   note that in the classic definition of the bishop - phelps - bollobs property the number @xmath62 depends only of @xmath39 . 
   so what happen if we ask for more in the definition of the strong bishop - phelps - bollobs property ? as we will see below , when we put @xmath129 to depends only of @xmath39 we get negative results . however , we use them to get examples of pairs of banach spaces that fail the sbpbp when the domain space is reflexive or when the range space has infinite dimension . 
   just to help to make reference we put a name of it . 
   we say that a pair of banach spaces @xmath4 has the _ uniform strong bishop - phelps - bollobs property _ 
   ( uniform sbpbp , for short ) if given @xmath39 , there exists @xmath40 such that whenever @xmath41 and @xmath15 are such that @xmath42 there is @xmath36 such that @xmath86    we observe that the kim - lee theorem says that a banach space @xmath0 is uniformly convex if and only if the pair @xmath60 has the uniform sbpbp where @xmath130 or @xmath131 . note also that if the pair @xmath4 satisfies the uniform sbpbp then the pair @xmath4 satisfies the bpbp . 
   the first thing that we notice is that if @xmath4 has the uniform sbpbp for some banach space @xmath1 , then @xmath0 must be uniformly convex .      given @xmath132 , consider @xmath40 the positive real number that satisfies the uniform sbpbp for the pair @xmath4 . 
   we prove that the pair @xmath60 has the uniform sbpbp with the same @xmath133 . 
   indeed , let @xmath57 and @xmath15 be such that @xmath134 let @xmath125 and define @xmath14 by @xmath135 for all @xmath136 . 
   so @xmath137 and @xmath138 since the pair @xmath4 has the uniform sbpbp with @xmath133 , there exists @xmath36 such that @xmath139 and @xmath140 . 
   since @xmath141 the proof is complete .          what about the reciprocal of corollary [ corsbpbp ] ? 
   the first thing that come to mind , since every hilbert space is uniformly convex , is to assume that the domain space @xmath0 is a hilbert space and try to find some banach space @xmath1 such that the pair @xmath4 satisfies the property . 
   but even in the simplest situation the result fails as we may see in the following example . 
   [ exsbpbp ] this example works for both real and complex cases . for a given @xmath39 , suppose that there exists @xmath40 satisfying the uniform sbpbp for the pair @xmath144 . 
   let @xmath145 be defined by @xmath146 for every @xmath147 . 
   for every @xmath148 , we have @xmath149 since @xmath150 , we obtain @xmath151 . 
   moreover , @xmath152 we prove now that every @xmath153 such that @xmath154 assumes the form @xmath155 for @xmath156 . 
   indeed , since @xmath157 and @xmath154 , we have @xmath158 . 
   since @xmath159 , we have @xmath160 and @xmath161 with @xmath156 . in summary 
   , we have a unit operator @xmath17 and a unit vector @xmath162 satisfying @xmath163 but if @xmath17 attains its norm at some point @xmath164 then @xmath165 with @xmath156 . 
   this contradicts the assumption that the pair @xmath144 has the uniform sbpbp since @xmath166 is far from @xmath162 in view of the fact that @xmath167 . 
   this shows that the pair @xmath168 fails to have the uniform sbpbp for @xmath130 or @xmath131 . 
   now what if we add on the hypothesis that both @xmath0 and @xmath1 are hilbert spaces ? 
   the answer for this question is still no as we can see below . 
   [ sbpbp8 ] this example works for both real and complex cases . for a given @xmath39 , suppose that there exists @xmath40 satisfying the uniform sbpbp for the pair @xmath169 . 
   define @xmath170 by @xmath171 for every @xmath147 . 
   then for every @xmath148 , @xmath172 and since @xmath173 , we have @xmath151 . also , we see that @xmath174 . now , 
   if @xmath153 is such that @xmath175 , then @xmath176 and since @xmath177 , we get @xmath178 which implies that @xmath160 and using again that @xmath159 , we obtain @xmath179 with @xmath156 and so @xmath180 which contradicts the hypothesis that the pair @xmath169 has the uniform sbpbp .      [ paramiguel1 ] let @xmath181 and @xmath182 be the space @xmath183 endowed with the @xmath23-norm and the @xmath184-norm , respectively , with @xmath185 ( or @xmath186 ) . given @xmath187 , there exists a bounded linear operator @xmath188 with @xmath189 such that      let @xmath187 and @xmath185 . 
   define @xmath188 by @xmath194 for every @xmath195 . 
   if @xmath196 , since @xmath197 , we get @xmath198 which implies that @xmath199 . 
   since @xmath200 , we have @xmath189 . 
   now , let @xmath201 be such that @xmath202 . 
   we prove that @xmath179 with @xmath156 . 
   indeed , the equality @xmath203 implies that @xmath204 and since @xmath205 , we do the difference between these two equalities to get @xmath206 since @xmath197 and @xmath207 , @xmath208 and @xmath209 . 
   because of the above equality , we get that @xmath210 . but 
   @xmath211 which implies that @xmath212 thus @xmath160 and then @xmath179 with @xmath156 as desired . 
   so if @xmath191 is such that @xmath192 , then @xmath213 which completes the proof .    as a consequence of this last result , we get that all the pairs @xmath214 fail the uniform sbpbp for @xmath185 for @xmath130 or @xmath131 . 
   now we study the pair @xmath215 in the real case when we put the sum norm on the range space . 
   unfortunatly , we can construct a bounded linear operator to get the same contradiction as before . 
   let @xmath216 and @xmath142 be the banach spaces @xmath217 endowed with the @xmath218-norm and the sum - norm , respectively . 
   given @xmath187 , there exists a bounded linear operator @xmath219 with @xmath189 such that      let @xmath187 . 
   define @xmath219 by @xmath223 for all @xmath224 . 
   then @xmath225 so @xmath226 and @xmath227 
   . also , we have @xmath189 . 
   indeed , let @xmath148 . 
   then @xmath228 recall that for every @xmath229 , we may write @xmath230 . with this in mind , if @xmath231 , then @xmath232 and if @xmath233 likewise we have @xmath234 + \frac{1}{2 } |\beta x - y| = \max \ { -\beta x , -y \ } 
   \leq 1.\ ] ] since @xmath235 , we have @xmath189 . now 
   , suppose that @xmath153 is such that @xmath221 . 
   so , @xmath236 again , we have two cases . if @xmath237 , then @xmath238 which implies @xmath239 since @xmath240 , and if @xmath241 , then @xmath242 which implies @xmath243 . since @xmath244 
   , we obtain @xmath160 and @xmath245 . 
   this means that if @xmath246 attains its norm at some @xmath247 , then @xmath180 . 
   [ paramiguel2 ] let @xmath216 and @xmath182 be the space @xmath217 endowed with the 2-norm and the @xmath184-norm , respectively , with @xmath250 . 
   given @xmath187 , there exists @xmath251 with @xmath189 such that      let @xmath250 and define @xmath254 by @xmath255 for every @xmath147 . 
   first of all , note that @xmath256 i.e. , @xmath257 . 
   this shows that @xmath258 . 
   next , we show that @xmath259 and also that the only points which @xmath17 attains its norm are at @xmath260 and @xmath261 . 
   to do so , we study the norm of the operator @xmath17 by using the following compact set : @xmath262 by symmetry , the norm of @xmath17 is the maximum of @xmath263 with @xmath166 in @xmath264 . 
   let @xmath265 a point of @xmath264 such that @xmath17 attains its norm at @xmath266 , that is , @xmath267 . 
   we consider @xmath268 as the segment that connect @xmath269 with @xmath162 , @xmath270 as the segment that connect @xmath269 with @xmath271 and @xmath272 as the arc that connect @xmath162 with @xmath271 . 
   see figure [ fig : maximum ] . 
   it is enough to study the values of @xmath273 on the set @xmath274 since the operator @xmath17 attains its norm at elements of the sphere and @xmath275 . 
   we have @xmath276 with @xmath277 defined as @xmath278 and @xmath279 defined as @xmath280 . since @xmath281 for every @xmath250 , then @xmath282 . 
   ( on the other hand observe that if @xmath283 , then @xmath284 and if @xmath285 , then @xmath286 . ) 
   thus if @xmath287 , then either @xmath288 and then @xmath289 would be a critical point of @xmath290 in @xmath291 , where @xmath292 or @xmath293 and in this case @xmath294 would be a critical point of @xmath295 in @xmath296 , where @xmath297.\ ] ] but , as we will see in the next lines , these can not happen because @xmath298 and @xmath299 for all @xmath300 and then @xmath301 . 
   indeed , we consider first the case that @xmath302 for every @xmath302 , we get @xmath303.\ ] ] for @xmath302 , we have that @xmath304 and since @xmath305 for every @xmath306 on this interval , we obtain that @xmath307 for every @xmath308 . a simply change of the letter @xmath290 by @xmath295 and @xmath306 by @xmath309 implies that @xmath299 for every @xmath310 . 
   everything we did so far was to prove that @xmath311 and that @xmath17 attains its norm on @xmath264 only at @xmath312 and @xmath313 . 
   therefore , we may conclude that @xmath17 attains its maximum at @xmath260 and at @xmath261 . 
   in other words , we proved that @xmath314 . 
   now , for @xmath315 , define @xmath251 by @xmath316 for every @xmath147 . 
   note that @xmath317 and @xmath318 . since @xmath319 , then @xmath199 . also , using that @xmath314 and that @xmath320 , we have that @xmath321 . 
   this implies that if @xmath164 is such that @xmath192 , then @xmath322 and therefore @xmath180 as we wanted . 
   what about the case that @xmath325 and @xmath250 
   ? we will study the real case of this right now . 
   consider @xmath325 . 
   define @xmath326 by @xmath327 for every @xmath328 . then @xmath329 and @xmath330 . 
   also , if @xmath331 , then @xmath332 and so @xmath333 . 
   this implies that @xmath334 and @xmath335 , and therefore @xmath336 thus @xmath337 . 
   given @xmath315 , let @xmath338 be as in the proposition [ paramiguel2 ] with @xmath250 . 
   now , define @xmath339 by @xmath340 . 
   then @xmath341 . 
   also , @xmath342 and @xmath343 . 
   suppose that there exists @xmath344 such that @xmath345 . 
   then @xmath346 and then , as we can see in the proof of the proposition [ paramiguel2 ] , @xmath166 must be equals to @xmath271 or @xmath347 . in both cases 
   , we have that @xmath348 . 
   we just have proved the following result . 
   suppose that there exists @xmath40 that depends only of a given @xmath39 satisfying the property . 
   let @xmath352 and @xmath52 be such that @xmath353 for @xmath354 . 
   define @xmath355 by @xmath356 for all @xmath136 . then @xmath357 and @xmath358 . 
   moreover , since @xmath359 , we have that @xmath259 . 
   this shows that @xmath151 . 
   therefore , there exists @xmath360 such that @xmath361 and @xmath362 . since @xmath363 and @xmath364 
   , we have that @xmath365 . on the other hand , since @xmath366 we get a contradiction , since @xmath367    it is a consequence of the fact that the pair @xmath144 fails the uniform sbpbp that the pair @xmath368)$ ] also fails this property . 
   indeed , consider @xmath369 $ ] positive functions defined on @xmath370 $ ] such that @xmath371 , @xmath372 $ ] and @xmath373 $ ] . 
   define @xmath374 $ ] by @xmath375 for all @xmath376 $ ] and @xmath377 . 
   then @xmath378 is linear . 
   since for @xmath379 we have @xmath380 and for @xmath381 $ ] we have @xmath382 , we get that @xmath383 } | \varphi(x , y)(t)| = \max \ { |x| , |y| \ } = \|(x , y)\|_{\infty}\ ] ] for all @xmath377 , using the fact that @xmath371 . 
   this shows that @xmath378 is a linear isometry between @xmath384 and the closed subspace of @xmath385 $ ] generated by @xmath386 and @xmath387 . 
   now suppose by contradiction that the pair @xmath368)$ ] has the uniform sbpbp . 
   then there exists @xmath40 for all @xmath132 , the modulus of the uniform sbpbp for this pair . 
   let @xmath388 with @xmath151 and @xmath389 be such that @xmath390 define @xmath391 $ ] by @xmath392 . 
   then for all @xmath148 , we have that @xmath393 and since @xmath378 is an isometry , we get that @xmath394 which implies that @xmath395 . also , @xmath396 so , there exists @xmath397 such that @xmath398 and @xmath399 . since @xmath400 , we just have proved that the pair @xmath144 also has the uniform sbpbp which contradicts example [ exsbpbp ] and proposition [ sbpbp1 ] . 
   next we show that if @xmath1 is a @xmath218-dimensional banach space , then the pair @xmath401 does not have the uniform sbpbp . to do so 
   , we use the existence of the auerbach base for a finite dimensional banach space ( see , for example , ( * ? ? ? 
   * proposition 20.21 ) ) . 
   let @xmath1 be an @xmath402-dimensional banach space . 
   then there are elements @xmath403 of @xmath1 and @xmath404 of @xmath405 such that @xmath406 for all @xmath407 and @xmath408 for @xmath409 . 
   in fact , @xmath410 is a base of @xmath1 called the auerbach base of @xmath1 . 
   let @xmath411 and @xmath412 satisfying @xmath406 for @xmath413 and @xmath408 for @xmath354 . since @xmath414 is a base for @xmath1 , every @xmath415 has an expression in terms of @xmath416 and @xmath417 given by @xmath418 . 
   given @xmath187 , define the continuous linear operator @xmath419 by @xmath420 for all @xmath421 . 
   then for all @xmath422 , we have that @xmath423 then @xmath199 . also , 
   note that @xmath424 . 
   so @xmath189 . 
   now let @xmath125 be such that @xmath425 . 
   then , using that @xmath426 we get that @xmath427 and therefore @xmath428 . 
   finally , if the pair @xmath401 has the uniform sbpbp , there exists @xmath40 satisfying the property . 
   if we put @xmath429 , there exists an operator @xmath430 such that @xmath151 , @xmath163 and for all @xmath125 which satisfies @xmath431 is such that @xmath432 . 
   this is a contradiction and the pair @xmath401 fails the uniform sbpbp . 
   it is clear but it is worth mentioning that if the pair @xmath4 has the uniform sbpbp , then the pair @xmath433 also has this property for all closed subspace @xmath434 of @xmath1 . 
   this implies , by the above proposition , that the pair @xmath401 fails the uniform sbpbp for all @xmath402-dimensional finite space @xmath1 . 
   anyway , just for curiosity , if @xmath435 , the proof of proposition [ yy ] works as well in this situation . 
   indeed , for @xmath187 we define @xmath436 by @xmath437 for all @xmath415 , where @xmath438 and @xmath439 is given by the auerbach basis . 
   then @xmath440 for @xmath441 and @xmath442 . to prove that @xmath199 , we add and subtract @xmath443 in @xmath444 where @xmath422 , to get @xmath445 . 
   now , it is clear that if @xmath246 attains its norm at some @xmath125 then @xmath446 for all @xmath441 . 
   if @xmath1 is a banach space which contains strictly convex @xmath218-dimensional subspaces , then there exists a uniformly convex banach space @xmath0 such that the pair @xmath4 fails the uniform sbpbp . 
   indeed , let @xmath434 be a subspace of @xmath1 such that @xmath434 is stricly convex and @xmath447 . 
   then @xmath448 is uniformly convex , since @xmath434 is finite dimensional . by proposition [ yy ] 
   , the pair @xmath433 fails the uniform sbpbp and by the above observation the pair @xmath4 can not have this property . 
   next we use the negatives results that we got so far about the uniform sbpbp to get examples of pairs @xmath4 that fail the strong bishop - phelps - bollobs prooperty . in remark 
   [ sbpbp5 ] we noted that if @xmath0 is not reflexive , then the pair @xmath4 does not have the sbpbp . in what follows we get examples of reflexive banach spaces @xmath0 such that the pair @xmath4 fails the sbpbp . 
   we also present a complete characterization for the pairs @xmath5 concerning this property by showing that there are cases that these pairs satisfy the property and other cases not ( see theorem [ sbpbp9 ] ) . 
   first of all we use the fact , which is showed in the next remark , that the pair @xmath449 fails the uniform sbpbp to get examples of banach spaces @xmath1 such that the pair @xmath450 fails the sbpbp ( see corollary [ sbpbp7 ] ) .    [ sbpbp4 ] by corollary [ sbpbp3 ] , the pair @xmath451 has the sbpbp if @xmath452 . 
   but in the case of the uniform sbpbp , we get a negative result . 
   we note that the pair @xmath449 fails the uniform sbpbp . indeed , 
   suppose by contradiction that this pair satisfies the property . 
   then given @xmath39 there exists @xmath40 such that whenever @xmath453 and @xmath454 are such that @xmath455 , there is @xmath456 such that @xmath139 and @xmath457 . 
   since the pair @xmath169 fails the uniform sbpbp , there exists some @xmath87 , a norm one linear operator @xmath458 and a norm one vector @xmath459 with @xmath460 such that there is no point @xmath461 such that @xmath462 and @xmath463 . 
   let @xmath464 be the projection on the first two coordinates , i.e. , @xmath465 for all @xmath466 . then @xmath467 . 
   define @xmath468 by @xmath469 . 
   then @xmath470 . let @xmath471 . 
   we have that @xmath472 then there exists @xmath473 such that @xmath474 and @xmath475 . since @xmath476 we get that @xmath477 . on the other hand , @xmath478 . 
   this is a contradiction and then the pair @xmath449 fails the sbpbp as desired . 
   the next theorem connects both sbpbp and uniform sbpbp in order to get examples of pairs of banach spaces @xmath4 that not satisfy the sbpbp by using the examples of the pairs that fail the uniform sbpbp . 
   suppose that the pair @xmath4 fails the uniform sbpbp . 
   then for each @xmath88 , there are @xmath480 , @xmath481 and @xmath89 with @xmath482 such that whenever @xmath124 satisfies @xmath483 we have @xmath484 . 
   define @xmath485 by @xmath486 for every @xmath487 . 
   since @xmath488 for all @xmath88 , we get that @xmath259 . on the other hand , 
   @xmath489 so @xmath151 . we suppose that there is @xmath490 such that the pair @xmath491 has the sbpbp with this constant . 
   denote by @xmath492 the element of @xmath493 such that in the @xmath402-th position is @xmath166 and in the rest is zero for all @xmath494 . 
   we take @xmath88 to be such that @xmath495 observe that @xmath496 thus there is @xmath497 such that @xmath498 by the second inequality , we get that @xmath499 for all @xmath500 and @xmath501 . moreover , by the equality @xmath502 we get that @xmath503 since @xmath504 for all @xmath500 . 
   so @xmath505 . 
   this contradicts the beggining of the proof and we conclude that the pair @xmath491 fails the sbpbp whenever @xmath4 fails the uniform sbpbp . 
   * @xmath144 , * @xmath169 , * @xmath510 for @xmath185 . 
   * @xmath215 , * @xmath511 , for @xmath250 . 
   * @xmath512 for @xmath324 . * @xmath513 for @xmath325 and @xmath350 . 
   * @xmath368)$ ] , * @xmath401 , where @xmath514 . 
   \(i ) by the pitt s theorem every bounded linear operator from @xmath56 into @xmath520 with @xmath519 is compact . by theorem [ sbpbp2 ] 
   the pair @xmath5 has the sbpbp since @xmath56 is uniformly convex for @xmath521 . 
   \(ii ) let @xmath132 and @xmath185 . 
   consider @xmath56 and @xmath520 as the banach spaces @xmath522 and @xmath523 , respectively . 
   for each @xmath88 define @xmath524 by @xmath525 for each @xmath195 . 
   let @xmath526 and let @xmath527 be defined by @xmath528 for each @xmath529 we have that @xmath530 and then @xmath531 so @xmath259 . we consider the vectors @xmath532 thus we get that @xmath533 . 
   so @xmath151 . 
   suppose that there exists @xmath84 such that the pair @xmath5 has the sbpbp . 
   let @xmath88 be such that @xmath534 . 
   so since @xmath535 and @xmath536 there exists @xmath537 such that @xmath538 next we claim that @xmath539 for all @xmath540 . 
   indeed , suppose that there exists some @xmath541 such that @xmath542 . 
   thus @xmath543 which is a contradiction . 
   then @xmath539 for all @xmath540 . 
   because of that , we have @xmath544 this new contradiction shows that the pair @xmath5 fails the sbpbp for @xmath185 
   .    * acknowledgement . 
   * this article is based on parts of my ph.d . 
   thesis at universidad de valencia and i would like to thank domingo garca , manuel maestre and miguel martn for their wonderful supervision on this work .
summary: in this paper we introduce the strong bishop - phelps - bollobs property ( sbpbp ) for bounded linear operators between two banach spaces @xmath0 and @xmath1 . 
   this property is motivated by a kim - lee result which states , under our notation , that a banach space @xmath0 is uniformly convex if and only if the pair @xmath2 satisfies the sbpbp . 
   positive results of pairs of banach spaces @xmath3 satisfying this property are given and concrete pairs of banach spaces @xmath4 failing it are exhibited . a complete characterization of the sbpbp for the pairs @xmath5 is also provided .