In this paper, we study weak stabi
lity properties of an
ε-isometry defined on a wedge
W of a Banach space
X, instead of the who
le space
X . As a resu
lt, we show that if
lsi1" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si1.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7a524f70cecbd3cc748ac79ec77e35f3" title="Click to view the MathML source">f:W→Ylass="mathContainer hidden">lass="mathCode"> is an
ε -isometry with
lsi2" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si2.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=6f47966b498b6841ce016481d04f6af7" title="Click to view the MathML source">f(0)=0lass="mathContainer hidden">lass="mathCode"> for some Banach space
Y , then there exists a
lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=5c57248395052000bfb911201e864cd8" title="Click to view the MathML source">w⁎lass="mathContainer hidden">lass="mathCode">-compact abso
lute
ly convex set
lsi120" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si120.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7e57573c133b3d8f1c31dd771d679c87" title="Click to view the MathML source">B⊂BX⁎lass="mathContainer hidden">lass="mathCode"> satisfying that (a)
lsi35" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si35.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=16ce1b3c3d9f85ae8681baf4a82dc868" title="Click to view the MathML source">p(x)≡supx⁎∈B〈x⁎,x〉=‖x‖lass="mathContainer hidden">lass="mathCode"> for a
ll lsi6" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si6.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=3c8ed623012adfb896c42e4a21c5b18e" title="Click to view the MathML source">x∈W∪−Wlass="mathContainer hidden">lass="mathCode">; and (b)
for every
lsi123" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si123.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=9e254289ddf1e499e99b4e71342241dd" title="Click to view the MathML source">x⁎∈Blass="mathContainer hidden">lass="mathCode">, there is
lsi124" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si124.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=73042084f40a79f2c1c495ad3213832e" title="Click to view the MathML source">ϕ∈BY⁎lass="mathContainer hidden">lass="mathCode"> so that
lass="
formu
la" id="fm0010">
This is a genera
lization of a recent resu
lt so ca
lled a universa
l theorem
for stabi
lity of
ε-isometries (but the proof is more technica
l). As its app
lication, we prove that if the
ε-isometry
f is defined on the positive cone
W of a
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M -space with a strong unit (in particu
lar,
lsi11" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si11.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=424b3780cc10fc1f3af8c925cc74f1ed" title="Click to view the MathML source">ℓ∞(Γ)lass="mathContainer hidden">lass="mathCode">, and
lsi12" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si12.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7252d66712a7bc5e77f298112e3d0070" title="Click to view the MathML source">L∞(μ)lass="mathContainer hidden">lass="mathCode"> for a finite measure
μ), then we can choose the set
B to be
lsi13" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si13.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=7721dd7fded495ab9269ecb107fd8059" title="Click to view the MathML source">BX⁎lass="mathContainer hidden">lass="mathCode">; the c
losed unit ba
ll of the dua
l lsi14" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si14.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=1e60c76f543eadbf805bc6a615373ff7" title="Click to view the MathML source">X⁎lass="mathContainer hidden">lass="mathCode">; and further show that
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lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=5c57248395052000bfb911201e864cd8" title="Click to view the MathML source">w⁎lass="mathContainer hidden">lass="mathCode">-to-
lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1500760X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1500760X&_rdoc=1&_issn=0022247X&md5=5c57248395052000bfb911201e864cd8" title="Click to view the MathML source">w⁎lass="mathContainer hidden">lass="mathCode"> continuous
ly isometric to a subspace of
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