On Lp Space Formed by Real-Valued Partial Functions

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On Lp Space Formed by Real-Valued Partial Functions

This article is the continuation of [31]. We define the set of Lp integrable functions - the set of all partial functions whose absolute value raised to the p-th power is integrable. We show that Lp integrable functions form the Lp space. We also prove Minkowski's inequality, Hölder's inequality and that Lp space is Banach space ([15], [27]).

[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[4] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.

[5] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.

[6] Józef Białas. The s-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.

[7] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.

[8] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.

[9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

[10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[13] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006, doi:10.2478/v10037-006-0008-x.

[14] Noboru Endou, Yasunari Shidama, and Keiko Narita. Egoroff's theorem. Formalized Mathematics, 16(1):57-63, 2008, doi:10.2478/v10037-008-0009-z.

[15] P. R. Halmos. Measure Theory. Springer-Verlag, 1987.

[16] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.

[17] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.

[18] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.

[19] Keiko Narita, Noboru Endou, and Yasunari Shidama. Integral of complex-valued measurable function. Formalized Mathematics, 16(4):319-324, 2008, doi:10.2478/v10037-008-0039-6.

[20] Keiko Narita, Noboru Endou, and Yasunari Shidama. Lebesgue's convergence theorem of complex-valued function. Formalized Mathematics, 17(2):137-145, 2009, doi: 10.2478/v10037-009-0015-9.

[21] Andrzej Nędzusiak. s-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.

[22] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.

[23] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.

[24] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.

[25] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.

[26] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.

[27] Walter Rudin. Real and Complex Analysis. Mc Graw-Hill, Inc., 1974.

[28] Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006, doi:10.2478/v10037-006-0018-8.

[29] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

[30] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[31] Yasushige Watase, Noboru Endou, and Yasunari Shidama. On L1 space formed by real-valued partial functions. Formalized Mathematics, 16(4):361-369, 2008, doi:10.2478/v10037-008-0044-9.

[32] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

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