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**Read Online or Download An Introduction to the Theory of Real Functions PDF**

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**Extra resources for An Introduction to the Theory of Real Functions**

**Sample text**

I=1 j=1 Assume now that l1 f1 , l2 f1 , . . , lm f1 , l1 f2 , l2 f2 , . . , lm f2 , . . , l1 fn , l2 fn , . . e. there exist a nontrivial linear combination n m dij (lj fi ) = e◦ i=1 j=1 lj then ni=1 ci fi = e◦ . Since the fi form a and not all dij = e◦ . Put ci = m j=1 dij basis and are thus linearly independent we must have ci = e◦ for all 1 ≤ i ≤ n. However, lj = e◦ implies that all dij = e◦ since l1 , l2 , . . , lm form a basis the equality m j=1 dij which contradicts the assumption.

Since K[x] forms a ring and K is a field, K[x] also forms a vector space over K. Example 121 Let K be a field, n ∈ IN and consider the subset Pn of K[x] defined by Pn = {f (x) ∈ K[x]| deg(f ) ≤ n} . Since addition of polynomials and multiplication by constants do not increase the degree, Pn is closed under addition and multiplication by scalars from K and is thus a vector space over K. The example of C being a vector space over IR can be generalized to arbitrary extension fields. Example 122 Let (K, ◦, ) be a field and let L ⊇ K be an extension field of K.

Definition 90 (Generator, principal ideal) Let (R, ◦, ) be a commutative ring and let I be an ideal. If there exist elements g1 , . . , gl such that I = {(g1 r1 ) ◦ · · · ◦ (gl rl )|r1 , . . rl ∈ R} then I is generated by g1 , . . , gl written I = (g1 , . . , gl ). If there exists a single element g ∈ I such that I = (g) = {g r|r ∈ R} then I is called a principal ideal. In this case, I is the ideal generated by g and g is called the generator of I. The ring R is called a principal ideal domain (PID) if every ideal is a principal ideal.