Law of Cosines for C .... Remmert Section 1.3, Ch. 0 ....

[Math Amateur]

I am reading Reinhold Remmert's book "Theory of Complex Functions" ...

I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.3: Scalar Product and Absolute Value ... ...

I need help in order to fully understand Remmert's derivation of the Law of Cosines for \(\displaystyle \mathbb{C}\)

The relevant part of Remmert's section on Scalar Product and Absolute Value reads as follows:View attachment 8548In the above text from Remmert we read the following:

" ... ... From the Cauchy-Schwarz Inequality it follows that

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \) for all \(\displaystyle w,z \in \mathbb{C}^{ \times } \)According to (non-trivial) results of calculus, for each \(\displaystyle w,z \in \mathbb{C}^{ \times }\) therefore a unique real number \(\displaystyle \phi\) with \(\displaystyle 0 \le \phi \le \pi\), exists satisfying \(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ... "
Can someone please demonstrate formally and rigorously exactly how ...

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \)

implies

\(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ...
Help will be appreciated ...

Peter========================================================================================It may help MHB readers of the above post to have access to the start of Remmert's Section 1.3 as it will help with the context and notation of the post ... so I am providing access to the same ... as follows:View attachment 8549Hope that helps ...

Peter

 

[Opalg]

In the above text from Remmert we read the following:

" ... ... From the Cauchy-Schwarz Inequality it follows that

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \) for all \(\displaystyle w,z \in \mathbb{C}^{ \times } \)

According to (non-trivial) results of calculus, for each \(\displaystyle w,z \in \mathbb{C}^{ \times }\) therefore a unique real number \(\displaystyle \phi\) with \(\displaystyle 0 \le \phi \le \pi\), exists satisfying

\(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ... "

Can someone please demonstrate formally and rigorously exactly how ...

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \)

implies

\(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ...

On the interval $[0,\pi]$, $\cos\phi$ is a strictly decreasing function. As $\phi$ goes from $0$ to $\pi$, $\cos\phi$ decreases from $1$ to $-1$, taking each value between $1$ and $-1$ exactly once. Therefore, given a number $x$ with $-1\leqslant x\leqslant1$, there will be a unique $\phi\in [0,\pi]$ such that $\cos\phi = x$.

As Remmert indicates, that is a non-trivial result from calculus. To demonstrate it formally and rigorously would require going back through the whole development of real-variable analysis.

 

[math771]

Sure, I can help explain how -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 implies \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }.

First, let's start by defining the scalar product and absolute value in \mathbb{C}. The scalar product of two complex numbers w and z is defined as \langle w, z \rangle = \text{Re} (w \overline{z}), where \text{Re}(w) is the real part of w and \overline{z} is the complex conjugate of z. The absolute value of a complex number w is defined as \mid w \mid = \sqrt{\langle w, w \rangle}.

Now, using the Cauchy-Schwarz Inequality, we have -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 for all w,z \in \mathbb{C}^{ \times }. This inequality tells us that the scalar product of two complex numbers w and z is bounded between -\mid w \mid \mid z \mid and \mid w \mid \mid z \mid. In other words, the scalar product of w and z lies on the unit circle in the complex plane.

Next, we use the fact that for each w,z \in \mathbb{C}^{ \times }, there exists a unique real number \phi with 0 \le \phi \le \pi satisfying \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }. This is a result of calculus that we can take as given.

Now, since the scalar product of w and z lies on the unit circle, we can interpret \phi as the angle between w and z. And since \phi satisfies \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }, we have shown that the Law of Cosines for \mathbb{C} holds, which states that the cosine of the angle between two complex numbers w and