By Werner W.

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**Sample text**

As the fn are Bochner Integrable, we may find simple functions f̃n such that ∫E ‖fn − f̃n ‖d???? → 0. 19) because ∫E ‖f − f̃n ‖d???? ≤ ∫E ‖f − fn ‖d???? + ∫E ‖fn − f̃n ‖d???? ◽ and the theorem has been proved. 7 If f is Bochner integrable, then ‖∫E fd????‖ ≤ ∫E ‖f ‖d????. Proof: Let fn = Then, ∑n i=1 gi IEi (????) be a simple function with Ei ∩ Ej = ???? for i ≠ j. n || || || ||∑ || fn (????)d???? || = |||| gi ????(Ei )|||| ||∫ || || || || E || || i=1 || n ∑ ≤ ‖gi ‖????(Ei ) = i=1 ∫E ‖fn ‖d????. 19), || || || || || || || fd???? || ≤ || fd???? − fn d???? || + || fn d???? || ||∫ || ||∫ | | | | || ∫ ∫E || E || || E || || E || || || ≤ |||| fd???? − fn d???? |||| + ‖fn ‖d???? ∫E ||∫E || ∫E || || ≤ |||| fd???? − fn d???? |||| + ‖fn − f ‖d???? + ‖f ‖d???? ∫E ∫E ||∫E || ∫E and the result follows upon taking limits with respect to n.

E. ???? is denoted by ????∞ (E, ℬ, ????), for which we define ‖f ‖∞ = ess sup|f (s)| s∈E = inf{x ∈ ℝ ∶ ????(s ∶ |f (s)| > x) = 0}. 4) Of course, the notation ‖ ⋅ ‖p suggests that we are working with a norm. 4. Somewhat more problematic is showing that ‖f ‖p = 0 means that f = 0. e. ???? in such an instance. To bypass this stumbling block, we must adopt the VECTOR AND FUNCTION SPACES 27 convention that functions differing only on a set of ???? measure 0 are identified as being the same function. That is, we define the equivalence relation ∼ by f ∼ g if ????{x ∶ f (x) ≠ g(x)} = 0 and focus on the quotient space ????p (E, ℬ, ????)∕ ∼ of equivalence classes.

The point here is that if ‖ ⋅ ‖1 and ‖ ⋅ ‖2 are equivalent, the metric spaces induced by the two norms have the same “topological” properties; for instance, the classes of open sets are the same, if one is complete then the other is also complete, etc. Thus, one can replace a particular norm with another that is equivalent, but possibly more mathematically tractable, if such a thing can be found. This is not so difficult in finite dimensions in view of the following theorem. 16 Let ???? be a finite-dimensional normed space.