A FEW WORDS ON SPACES, VECTORS AND FUNCTIONS

A vector space means a set V of elements x,y. that form an Abelian group and can be added together and multiplied by numbers z = ax + f3y thus producing zeV. The multiplication (a, /3 are, in general, complex numbers) satisfies the usual rules (the group is Abelian, because x + y = y + x)  

Example 2. Integers with real muUipUers. If, in the previous example, we admitted a, /3 to be real, the multiplication of integers jc, y by real numbers would give real numbers (not necessarily integers). Therefore, in this case x,y.do not represent any vector space. 

Note that if onty the positive vector components were allowed, they would not form an Abelian group (no neutral element), and on top of this their addition (which might mean a subtraction of components, because a, j8 could be negative) could produce vectors with non-positive components. Thus vectors with all positive components do not form a vector space. 

Example 4. Functions. This example is important in the context of this book. This time the vectors have real components. Their addition means the addition of two functions /(jc) = fi x) + fzix). The multiplication means multiplication by a real number. The unit ( neutral ) function means / = 0, the inverse function to / is —/(jc). Therefore, the functions form an Abelian group. A few seconds are needed to show that the four axioms above are satisfied. Such functions form a vector space. 

Linear independence. A set of vectors is called a set of linearly independent vectors if no vector of the set can be pressed as a linear combination of the other vectors of the set. The number of linearly independent vectors in a vector space is called the dimension of the space. 

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The results of this section, even with their limitations, are the punch line of our story, the particularly beautiful goal promised in the preface. Now is a perfect time for the reader to take a few moments to reflect on the journey. We have studied a significant amount of mathematics, including approximations in vector spaces of functions, representations, invariance, isomorphism, irreducibility and tensor products. We have used some big theorems, such as the Stone-Weierstrass Theorem, Fubini s Theorem and the Spectral Theorem. Was it worth it And, putting aside any aesthetic pleasure the reader may have experienced, was it worth it from the experimental point of view In other words, are the predictions of this section worth the effort of building the mathematical machinery ... 

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