Banach space (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Banach space" in English language version.

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ams.org

Anderson & Schori 1969, p. 315. Anderson, R. D.; Schori, R. (1969). "Factors of infinite-dimensional manifolds" (PDF). Transactions of the American Mathematical Society. 142. American Mathematical Society (AMS): 315–330. doi:10.1090/s0002-9947-1969-0246327-5. ISSN 0002-9947.
Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4. Archived (PDF) from the original on 2022-10-09.

mathscinet.ams.org

Henderson 1969. Henderson, David W. (1969). "Infinite-dimensional manifolds are open subsets of Hilbert space". Bull. Amer. Math. Soc. 75 (4): 759–762. doi:10.1090/S0002-9904-1969-12276-7. MR 0247634.

amu.edu.pl

kielich.amu.edu.pl

Banach (1932, p. 182) Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see Banach (1932), pp. 11-12. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see Banach (1932), Th. 9 p. 185. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see also Banach (1932), p. 170 for metrizable $K$ and $L.$ Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
the question appears p. 238, §3 in Banach's book, Banach (1932). Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see p. 245 in Banach (1932). The homogeneity property is called "propriété (15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec $(L^{2}).$ possède la propriété (15)". Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.

archive.org

see Corollary 2, p. 11 in Diestel (1984). Diestel, Joseph (1984), Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261, ISBN 0-387-90859-5.
see p. 85 in Diestel (1984). Diestel, Joseph (1984), Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261, ISBN 0-387-90859-5.
see p. 201 in Diestel (1984). Diestel, Joseph (1984), Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261, ISBN 0-387-90859-5.

arxiv.org

Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
Rosenthal, Haskell P (1974). "A characterization of Banach spaces containing ℓ¹". Proc. Natl. Acad. Sci. U.S.A. 71 (6): 2411–2413. arXiv:math.FA/9210205. Bibcode:1974PNAS...71.2411R. doi:10.1073/pnas.71.6.2411. PMC 388466. PMID 16592162. Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in Dor, Leonard E (1975). "On sequences spanning a complex ℓ¹ space". Proc. Amer. Math. Soc. 47: 515–516. doi:10.1090/s0002-9939-1975-0358308-x.
see Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1995). "Banach spaces without local unconditional structure". Israel Journal of Mathematics. 89 (1–3): 205–226. arXiv:math/9306211. doi:10.1007/bf02808201. S2CID 5220304. and also Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1998). "Erratum to: Banach spaces without local unconditional structure". Israel Journal of Mathematics. 105: 85–92. arXiv:math/9607205. doi:10.1007/bf02780323. S2CID 18565676.

books.google.com

Bessaga & Pełczyński 1975, p. 189 Bessaga, C.; Pełczyński, A. (1975), Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, Warszawa: Panstwowe wyd. naukowe.
Aliprantis & Border 2006, p. 185. Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874.
C. Bessaga, A. Pełczyński (1975). Selected Topics in Infinite-Dimensional Topology. Panstwowe wyd. naukowe. pp. 177–230.

dml.cz

Odell, Edward W.; Rosenthal, Haskell P. (1975), "A double-dual characterization of separable Banach spaces containing ℓ¹" (PDF), Israel Journal of Mathematics, 20 (3–4): 375–384, doi:10.1007/bf02760341, S2CID 122391702, archived (PDF) from the original on 2022-10-09.

doi.org

Anderson & Schori 1969, p. 315. Anderson, R. D.; Schori, R. (1969). "Factors of infinite-dimensional manifolds" (PDF). Transactions of the American Mathematical Society. 142. American Mathematical Society (AMS): 315–330. doi:10.1090/s0002-9947-1969-0246327-5. ISSN 0002-9947.
Henderson 1969. Henderson, David W. (1969). "Infinite-dimensional manifolds are open subsets of Hilbert space". Bull. Amer. Math. Soc. 75 (4): 759–762. doi:10.1090/S0002-9904-1969-12276-7. MR 0247634.
Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4. Archived (PDF) from the original on 2022-10-09.
Eilenberg, Samuel (1942). "Banach Space Methods in Topology". Annals of Mathematics. 43 (3): 568–579. doi:10.2307/1968812. JSTOR 1968812.
Amir, Dan (1965). "On isomorphisms of continuous function spaces". Israel Journal of Mathematics. 3 (4): 205–210. doi:10.1007/bf03008398. S2CID 122294213.
Cambern, M. (1966). "A generalized Banach–Stone theorem". Proc. Amer. Math. Soc. 17 (2): 396–400. doi:10.1090/s0002-9939-1966-0196471-9. And Cambern, M. (1967). "On isomorphisms with small bound". Proc. Amer. Math. Soc. 18 (6): 1062–1066. doi:10.1090/s0002-9939-1967-0217580-2.
Cohen, H. B. (1975). "A bound-two isomorphism between $C(X)$ Banach spaces". Proc. Amer. Math. Soc. 50: 215–217. doi:10.1090/s0002-9939-1975-0380379-5.
R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998.
bishop, See E.; Phelps, R. (1961). "A proof that every Banach space is subreflexive". Bull. Amer. Math. Soc. 67: 97–98. doi:10.1090/s0002-9904-1961-10514-4.
Rosenthal, Haskell P (1974). "A characterization of Banach spaces containing ℓ¹". Proc. Natl. Acad. Sci. U.S.A. 71 (6): 2411–2413. arXiv:math.FA/9210205. Bibcode:1974PNAS...71.2411R. doi:10.1073/pnas.71.6.2411. PMC 388466. PMID 16592162. Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in Dor, Leonard E (1975). "On sequences spanning a complex ℓ¹ space". Proc. Amer. Math. Soc. 47: 515–516. doi:10.1090/s0002-9939-1975-0358308-x.
Odell, Edward W.; Rosenthal, Haskell P. (1975), "A double-dual characterization of separable Banach spaces containing ℓ¹" (PDF), Israel Journal of Mathematics, 20 (3–4): 375–384, doi:10.1007/bf02760341, S2CID 122391702, archived (PDF) from the original on 2022-10-09.
for more on pointwise compact subsets of the Baire class, see Bourgain, Jean; Fremlin, D. H.; Talagrand, Michel (1978), "Pointwise Compact Sets of Baire-Measurable Functions", Am. J. Math., 100 (4): 845–886, doi:10.2307/2373913, JSTOR 2373913.
see Enflo, P. (1973). "A counterexample to the approximation property in Banach spaces". Acta Math. 130: 309–317. doi:10.1007/bf02392270. S2CID 120530273.
Lindenstrauss, Joram; Tzafriri, Lior (1971). "On the complemented subspaces problem". Israel Journal of Mathematics. 9 (2): 263–269. doi:10.1007/BF02771592.
see Gowers, W. T. (1994). "A solution to Banach's hyperplane problem". Bull. London Math. Soc. 26 (6): 523–530. doi:10.1112/blms/26.6.523.
see Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1995). "Banach spaces without local unconditional structure". Israel Journal of Mathematics. 89 (1–3): 205–226. arXiv:math/9306211. doi:10.1007/bf02808201. S2CID 5220304. and also Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1998). "Erratum to: Banach spaces without local unconditional structure". Israel Journal of Mathematics. 105: 85–92. arXiv:math/9607205. doi:10.1007/bf02780323. S2CID 18565676.

ghostarchive.org

Conrad, Keith. "Equivalence of norms" (PDF). kconrad.math.uconn.edu. Archived (PDF) from the original on 2022-10-09. Retrieved September 7, 2020.
Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4. Archived (PDF) from the original on 2022-10-09.
Odell, Edward W.; Rosenthal, Haskell P. (1975), "A double-dual characterization of separable Banach spaces containing ℓ¹" (PDF), Israel Journal of Mathematics, 20 (3–4): 375–384, doi:10.1007/bf02760341, S2CID 122391702, archived (PDF) from the original on 2022-10-09.

harvard.edu

ui.adsabs.harvard.edu

R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998.
Rosenthal, Haskell P (1974). "A characterization of Banach spaces containing ℓ¹". Proc. Natl. Acad. Sci. U.S.A. 71 (6): 2411–2413. arXiv:math.FA/9210205. Bibcode:1974PNAS...71.2411R. doi:10.1073/pnas.71.6.2411. PMC 388466. PMID 16592162. Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in Dor, Leonard E (1975). "On sequences spanning a complex ℓ¹ space". Proc. Amer. Math. Soc. 47: 515–516. doi:10.1090/s0002-9939-1975-0358308-x.

jstor.org

Eilenberg, Samuel (1942). "Banach Space Methods in Topology". Annals of Mathematics. 43 (3): 568–579. doi:10.2307/1968812. JSTOR 1968812.
for more on pointwise compact subsets of the Baire class, see Bourgain, Jean; Fremlin, D. H.; Talagrand, Michel (1978), "Pointwise Compact Sets of Baire-Measurable Functions", Am. J. Math., 100 (4): 845–886, doi:10.2307/2373913, JSTOR 2373913.

nih.gov

ncbi.nlm.nih.gov

R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998.
Rosenthal, Haskell P (1974). "A characterization of Banach spaces containing ℓ¹". Proc. Natl. Acad. Sci. U.S.A. 71 (6): 2411–2413. arXiv:math.FA/9210205. Bibcode:1974PNAS...71.2411R. doi:10.1073/pnas.71.6.2411. PMC 388466. PMID 16592162. Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in Dor, Leonard E (1975). "On sequences spanning a complex ℓ¹ space". Proc. Amer. Math. Soc. 47: 515–516. doi:10.1090/s0002-9939-1975-0358308-x.

pubmed.ncbi.nlm.nih.gov

R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998.
Rosenthal, Haskell P (1974). "A characterization of Banach spaces containing ℓ¹". Proc. Natl. Acad. Sci. U.S.A. 71 (6): 2411–2413. arXiv:math.FA/9210205. Bibcode:1974PNAS...71.2411R. doi:10.1073/pnas.71.6.2411. PMC 388466. PMID 16592162. Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in Dor, Leonard E (1975). "On sequences spanning a complex ℓ¹ space". Proc. Amer. Math. Soc. 47: 515–516. doi:10.1090/s0002-9939-1975-0358308-x.

qchu.wordpress.com

Qiaochu Yuan (June 23, 2012). "Banach spaces (and Lawvere metrics, and closed categories)". Annoying Precision.

semanticscholar.org

api.semanticscholar.org

Amir, Dan (1965). "On isomorphisms of continuous function spaces". Israel Journal of Mathematics. 3 (4): 205–210. doi:10.1007/bf03008398. S2CID 122294213.
Odell, Edward W.; Rosenthal, Haskell P. (1975), "A double-dual characterization of separable Banach spaces containing ℓ¹" (PDF), Israel Journal of Mathematics, 20 (3–4): 375–384, doi:10.1007/bf02760341, S2CID 122391702, archived (PDF) from the original on 2022-10-09.
see Enflo, P. (1973). "A counterexample to the approximation property in Banach spaces". Acta Math. 130: 309–317. doi:10.1007/bf02392270. S2CID 120530273.
see Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1995). "Banach spaces without local unconditional structure". Israel Journal of Mathematics. 89 (1–3): 205–226. arXiv:math/9306211. doi:10.1007/bf02808201. S2CID 5220304. and also Komorowski, Ryszard A.; Tomczak-Jaegermann, Nicole (1998). "Erratum to: Banach spaces without local unconditional structure". Israel Journal of Mathematics. 105: 85–92. arXiv:math/9607205. doi:10.1007/bf02780323. S2CID 18565676.

uconn.edu

kconrad.math.uconn.edu

Conrad, Keith. "Equivalence of norms" (PDF). kconrad.math.uconn.edu. Archived (PDF) from the original on 2022-10-09. Retrieved September 7, 2020.

web.archive.org

Banach (1932, p. 182) Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see Banach (1932), pp. 11-12. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see Banach (1932), Th. 9 p. 185. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see also Banach (1932), p. 170 for metrizable $K$ and $L.$ Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
the question appears p. 238, §3 in Banach's book, Banach (1932). Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see p. 245 in Banach (1932). The homogeneity property is called "propriété (15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec $(L^{2}).$ possède la propriété (15)". Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.

worldcat.org

search.worldcat.org

The normed space $(\mathbb {R} ,|\cdot |)$ is a Banach space where the absolute value is a norm on the real line $\mathbb {R}$ that induces the usual Euclidean topology on $\mathbb {R} .$ Define a metric $D:\mathbb {R} \times \mathbb {R} \to \mathbb {R}$ on $\mathbb {R}$ by $D(x,y)=|\arctan(x)-\arctan(y)|$ for all $x,y\in \mathbb {R} .$ Just like $|\cdot |$ 's induced metric, the metric $D$ also induces the usual Euclidean topology on $\mathbb {R} .$ However, $D$ is not a complete metric because the sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ defined by $x_{i}:=i$ is a $D$ -Cauchy sequence but it does not converge to any point of $\mathbb {R} .$ As a consequence of not converging, this $D$ -Cauchy sequence cannot be a Cauchy sequence in $(\mathbb {R} ,|\cdot |)$ (that is, it is not a Cauchy sequence with respect to the norm $|\cdot |$ ) because if it was $|\cdot |$ -Cauchy, then the fact that $(\mathbb {R} ,|\cdot |)$ is a Banach space would imply that it converges (a contradiction).Narici & Beckenstein 2011, pp. 47–51 Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Narici & Beckenstein 2011, p. 93. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Wilansky 2013, p. 29. Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
Anderson & Schori 1969, p. 315. Anderson, R. D.; Schori, R. (1969). "Factors of infinite-dimensional manifolds" (PDF). Transactions of the American Mathematical Society. 142. American Mathematical Society (AMS): 315–330. doi:10.1090/s0002-9947-1969-0246327-5. ISSN 0002-9947.
Aliprantis & Border 2006, p. 185. Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874.
Trèves 2006, p. 145. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Trèves 2006, pp. 166–173. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Narici & Beckenstein 2011, pp. 47–66. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Narici & Beckenstein 2011, pp. 47–51. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer & Wolff 1999, p. 35. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves 2006, pp. 57–69. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Trèves 2006, p. 201. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Narici & Beckenstein 2011, pp. 192–193. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.

zbmath.org

Banach (1932, p. 182) Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see Banach (1932), pp. 11-12. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see Banach (1932), Th. 9 p. 185. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see also Banach (1932), p. 170 for metrizable $K$ and $L.$ Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
the question appears p. 238, §3 in Banach's book, Banach (1932). Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
see p. 245 in Banach (1932). The homogeneity property is called "propriété (15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec $(L^{2}).$ possède la propriété (15)". Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.