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• [2] S. Banach, Théorie des opérations linéaires, Warsaw, 1932.
• [3] Garrett Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), no. 2, 169–172. MR 1545873, https://doi.org/10.1215/S0012-7094-35-00115-6
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• [5] Frederick A. Ficken, Note on the existence of scalar products in normed linear spaces, Ann. of Math. (2) 45 (1944), 362–366. MR 10255, https://doi.org/10.2307/1969273
• [6] Robert Fortet, Remarques sur les espaces uniformément convexes, C. R. Acad. Sci. Paris 210 (1940), 497–499. MR 2007
• [7] R. Fortet, Remarques sur les espaces uniformément convexes, Bull. Soc. Math. France 69 (1941), 23–46 (French). MR 13225
• [8] M. Fréchet, Sur la notion de différentielle, J. Math. Pures Appl. vol. 16 (1937) pp. 233-250.
• [9] -, Sur la définition axiomatique d'une classe d'espaces vectoriels distanciés applicables vectoriellement sur l'espaces de Hilbert, Ann. of Math. vol. 36 (1935) pp. 705-718.
• [10] V. Gantmakher and V. Šmulian, Sur les espaces linéaires dont la sphere unitaire est faiblement compacte, C. R. (Doklady) Acad. Sci. URSS. vol. 17 (1937) pp. 91-94.
• [11] R. C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291–302. MR 12199
• [12] P. Jordan and J. Von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719–723. MR 1503247, https://doi.org/10.2307/1968653
• [13] H. Löwig, Komplexe Euklidische Räume von beliebiger endlicher oder transfiniter Dimensionzahl, Acta Litterarum Ac Scientiarum vol. 7 (1934-1935) pp. 1-33.
• [14] S. Mazur, Über convexe Mengen in linearen normierten Räumen, Studia Mathematica vol. 4 (1933) pp. 70-84.
• [15] -, Über schwache Konvergenz in den Räumen , Studia Mathematica vol. 4 (1933) pp. 128-133.
• [16] D. Milman, On some criteria for the regularity of spaces of the type , C. R. (Doklady) Acad. Sci. URSS. vol. 20 (1938) pp. 243-246.
• [17] B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math. J. 5 (1939), no. 2, 249–253. MR 1546121, https://doi.org/10.1215/S0012-7094-39-00522-3
• [18] B. D. Roberts, On the geometry of abstract vector spaces, Tôhoku Math. J. vol. 39 (1934) pp. 42-59.
• [19] V. Šmulian, On some geometrical properties of the unit sphere in the space of the type , Rec. Math. (Mat. Sbornik) N. S. vol. 48 (1938) pp. 90-94.
• [20] V. Šmulian, Sur la dérivabilité de la norme dans l’espace de Banach, C. R. (Doklady) Acad. Sci. URSS (N. S.) 27 (1940), 643–648 (French). MR 0002704
• [21] A. E. Taylor, The extension of linear functionals, Duke Math. J. 5 (1939), 538–547. MR 345
• [22] A. E. Taylor, Derivatives in the calculus, Amer. Math. Monthly 49 (1942), 631–642. MR 7016, https://doi.org/10.2307/2302570

Bibliography

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G. Ascoli, Sugli spazi lineari metrici e le loro varieta lineari, Annali di Matemática Pura ed Applicata (4) vol. 10 (1932) pp. 33-81. MR 1553181

[2]

S. Banach, Théorie des opérations linéaires, Warsaw, 1932.

[3]

G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. vol. 1 (1935) pp. 169-172. MR 1545873

[4]

J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. vol. 40 (1936) pp. 396-414. MR 1501880

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F. A. Ficken, Note on the existence of scalar products in normed linear spaces, Ann. of Math. vol. 45 (1944) pp. 362-366. MR 0010255 (5:270f)

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R. Fortet, Remarques sur les espaces uniformément convexes, C. R. Acad. Sci. Paris vol. 210 (1940) pp. 497-499. MR 0002007 (1:335c)

[7]

-, Remarques sur les espaces uniformément convexes, Bull. Soc. Math. France vol. 69 (1941) pp. 23-46. MR 0013225 (7:124f)

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M. Fréchet, Sur la notion de différentielle, J. Math. Pures Appl. vol. 16 (1937) pp. 233-250.

[9]

-, Sur la définition axiomatique d'une classe d'espaces vectoriels distanciés applicables vectoriellement sur l'espaces de Hilbert, Ann. of Math. vol. 36 (1935) pp. 705-718.

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V. Gantmakher and V. Šmulian, Sur les espaces linéaires dont la sphere unitaire est faiblement compacte, C. R. (Doklady) Acad. Sci. URSS. vol. 17 (1937) pp. 91-94.

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R. C. James, Orthogonality in normed linear spaces, Duke Math. J. vol. 12 (1945) pp. 291-302. MR 0012199 (6:273d)

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P. Jordan and J. von Neumann, Inner products in linear, metric spaces, Ann. of Math. vol. 36 (1935) pp. 719-723. MR 1503247

[13]

H. Löwig, Komplexe Euklidische Räume von beliebiger endlicher oder transfiniter Dimensionzahl, Acta Litterarum Ac Scientiarum vol. 7 (1934-1935) pp. 1-33.

[14]

S. Mazur, Über convexe Mengen in linearen normierten Räumen, Studia Mathematica vol. 4 (1933) pp. 70-84.

[15]

-, Über schwache Konvergenz in den Räumen , Studia Mathematica vol. 4 (1933) pp. 128-133.

[16]

D. Milman, On some criteria for the regularity of spaces of the type , C. R. (Doklady) Acad. Sci. URSS. vol. 20 (1938) pp. 243-246.

[17]

B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math. J. vol. 5 (1939) pp. 249-253. MR 1546121

[18]

B. D. Roberts, On the geometry of abstract vector spaces, Tôhoku Math. J. vol. 39 (1934) pp. 42-59.

[19]

V. Šmulian, On some geometrical properties of the unit sphere in the space of the type , Rec. Math. (Mat. Sbornik) N. S. vol. 48 (1938) pp. 90-94.

[20]

-, Sur la dérivabilité de la norme dans l'espace de Banach, C. R. (Doklady) Acad. Sci. URSS. vol. 27 (1940) pp. 643-648. MR 0002704 (2:102f)

[21]

A. E. Taylor, The extension of linear functionals, Duke Math. J. vol. 5 (1939) pp. 538-547. MR 0000345 (1:58b)

[22]

-, Derivatives in the calculus, Amer. Math. Monthly vol. 49 (1942) pp. 631-642. MR 0007016 (4:73a)