Banach Spaces: The Big Results (from Textbooks)

This post shows how the big theorems of Banach space theory are stated in four textbooks. It complements a previous post.

Principal references: Aliprantis and Border (2006), Royden and Fitzpatrick (2010), Megginson (1998), Holmes (1975). Also useful Brezis (2011), Lax (2014).

Figure 1: Holmes p 24.

Figure 2: Holmes p 95.

Compactness

Section numbers refer to Royden.

12.2 Tychonoff

Figure 3: Roden.

Figure 4: Aliprantis.

10.1 Arzela-Ascoli

Figure 5: Roden.

Foundations

10.2 Baire Category theorem

Figure 6: Roden. A hollow set is one whose closure has an empty interior.

Figure 7: Roden.

Figure 8: Application to continuous, nowhere differentiable functions. p. 133, Holmes.

Figure 9: Megginson.

Figure 10: Megginson uses Zabreiko’s lemma as a basis for the other three theorems.

13.4 Open Mapping

Figure 11: Royden.

Figure 12: Royden.

Figure 13: Aliprantis.

Figure 14: Megginson.

Figure 15: Megginson.

13.4 Closed Graph

Figure 16: Royden.

Figure 17: Aliprantis.

Figure 18: Megginson.

13.5 Uniform Boundedness / Banach Steinhaus

Figure 19: Royden.

Figure 20: Royden.

Figure 21: Aliprantis.

Figure 22: Royden.

Figure 23: Megginson.

14.2 Hahn-Banach

Figure 24: Royden.

Figure 25: Royden.

Figure 26: Royden.

Figure 27: Aliprantis.

Figure 28: Megginson.

Figure 29: Megginson.

Weak* Duality and Compactness

15.1 Banach Alaoglu (extension of Helley’s theorem)

Figure 30: Royden.

Figure 31: Royden.

15.2 Kakutani

Figure 32: Royden.

15.3 Goldstine

Royden.

Figure 33: Megginson.

15.3 Eberlein Šmulian

Figure 34: Royden.

Figure 35: Royden.

Weak compactness

Figure 36: Royden.

Bishop Phelps Aliprantis Ch 7 convexity

Figure 37: Aliprantis.

Figure 38: Aliprantis.

Figure 39: Megginson.

Figure 40: Megginson.

James (166 is Holmes (1975), S19 p.157)

Figure 41: Aliprantis.

Figure 42: Holmes.

Figure 43: Holmes.

Figure 44: Proximinal, Megginson.

Figure 45: Megginson.

Figure 46: Megginson.

8.3 Helley / weak sequential compactness

Figure 47: Royden.

Bounded sequences can fail to have strongly convergent subsequences, but we get a weak version - compactness regained.

Figure 48: Royden.

Convex Geometry

14.5 Mazur

Figure 49: Royden.

14.6 Krein-Milman (hyperplane sep in 14.5)

Figure 50: Royden.

Figure 51: Royden.

Figure 52: Royden.

Figure 53: Aliprantis.

Measure Theory and Representations

8.1 Riesz Representation (Aliprantis has whole chapter) functional to measure

Figure 54: Aliprantis.

Figure 55: Aliprantis.

18.4 Radon Nikoydm measure to function

Figure 56: Royden.

Figure 57: Aliprantis.

4.6 Vitali Convergence and UI

Figure 58: Royden.

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Miscellaneous

Figure 59: Some characterizations of finite dimensionality 13.3, Royden.

Figure 60: Royden.

Figure 61: 15.4 Non-metrizability, Royden.

A normed linear space is reflexive iff weak and weak* topologies in on \(X^*\) are the same p276.

References

Aliprantis, Charalambos D., and Kim C. Border, 2006, Infinite Dimensional Analysis: A Hitchhiker’s Guide. Third. (Springer Verlag).

Brezis, Haim, 2011, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer New York).

Holmes, Richard B., 1975, Geometric Functional Analysis and its Applications Graduate Texts in Mathematics (Springer New York).

Lax, Peter D., 2014, Functional analysis (John Wiley & Sons).

Megginson, Robert E., 1998, An Introduction to Banach Space Theory Graduate Texts in Mathematics (Springer New York).

Royden, Halsey, and Patrick Fitzpatrick, 2010, Real Analysis. Fourth. (Pearson).