By Arthur Wouk

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**Additional resources for A course of applied functional analysis**

**Sample text**

1 4 1 2 2 1 2 1 3 1 2 5 4 3 4 3 4 3 4 2. 3 6 5 6 5 6 6 5 5 6 8. k = { Pi } ; l = { Pi, P2 } ; m = { Pi, Pa } ; n = { P2, Pa } . 9. k = { P2, P a, P" } ; l = { Pi, P2, Pa } ; m = { Pi, P2, P 4 } . 10. l = { Pi, Pa, P2 } . 11. 1 2 3 4 5 6 7 2 3 4 5 6 7 1. 4 5 6 7 1 2 3 12. l = { P i, P2, Pa } ; m = { P", Ps, P6 } . II. Consider the following axiom system : 1. If l and m are any two distinct lines, they have at least one point in common. 2. If Pi and P2 are any two distinct points, they have at least one line through them.

To explain precisely how the test works, it will be helpful to introduce a few definitions and then to use the axiomatic systems of the preceding chapter to illustrate the test. , ............. introduced as Axiom Set 1 and Axiom Set 2 are "abstract systems" as long as the terms "point" and "line" are taken as undefined. As long as these terms remain undefined the axioms are open sentences. It is not until some meaning is given to the undefined terms that one may legitimately ask whether the axioms are }rue or false.

2. If Z is a line, then there exist at least three points on it. 3. If Z is a line, then there exists a point P not on it. 4. There exists at least one line. 5a. If Z is a line and P a point not on it, then there exists at least one line m through P with no point in common with Z. 5b. If Z is a line and P a point not on it, then there exists at most one line m through P with no point in common with l. Axio m Set 2 la. If P and Q are any two points, then there exists at least one line containing both P and Q.