1.6. A NORMED VECTOR SPACE Fp 15

and the first completely correct proof was due to Argand in 1806. For more on this theo-rem, you can google fundamental theorem of algebra and look at the interesting Wikipediaarticle on it. Proofs of this theorem usually involve the use of techniques from calculuseven though it is really a result in algebra. A proof and plausibility explanation is givenlater.

Recall the quadratic formula which gives solutions to ax2 +bx+ c = 0 which holds forany a,b,c ∈ C with a ̸= 0. This is also good to review from any good pre-calculus book.My book published withhttp://www.centerofmath.org/textbooks/pre calc/index.html(2012) has all of these elemen-tary considerations. Most are in my on line calculus text or Volume 1 of the one publishedby World Scientific.

1.5.2 The Complex ExponentialHere is a short review of the complex exponential.

It was shown above that every complex number is of the form r (cosθ + isinθ) wherer ≥ 0. Laying aside the zero complex number, this shows that every non zero complexnumber is of the form eα (cosβ + isinβ ) . We write this in the form eα+iβ . Having doneso, does it follow that the expression preserves the most important property of the functiont→ e(α+iβ )t for t real, that (

e(α+iβ )t)′

= (α + iβ )e(α+iβ )t?

By the definition just given which does not contradict the usual definition in case β = 0 andthe usual rules of differentiation in calculus,(

e(α+iβ )t)′

=(eαt (cos(β t)+ isin(β t))

)′= eαt [α (cos(β t)+ isin(β t))+(−β sin(β t)+ iβ cos(β t))]

Now consider the other side. From the definition it equals

(α + iβ )(eαt (cos(β t)+ isin(β t))

)= eαt [(α + iβ )(cos(β t)+ isin(β t))]

= eαt [α (cos(β t)+ isin(β t))+(−β sin(β t)+ iβ cos(β t))]

which is the same thing. This is of fundamental importance in differential equations. Itshows that there is no change in going from real to complex numbers for ω in the consid-eration of the problem y′ = ωy, y(0) = 1. The solution is always eωt . The formula justdiscussed, that

eα (cosβ + isinβ ) = eα+iβ

is Euler’s formula. He originally conceived of this formula by considering power series ofcos and sin and re arranging the order of the infinite sums.

1.6 A Normed Vector Space Fp

In this book F will denote either the complex numbers C or the real numbers R. For p apositive integer,

Fp ≡{(a1, · · · ,ap) : ak ∈ F

}