Baby Rudin Proof of Theorem 1.33 (e) - Triangle Inequality

[josueortega]

Hi everyone,

I have a question on Rudin's proof of Theorem 1.33 part e. Here he prove the following statement:

The absolute value of z+w is equal or smaller than the absolute value of z plus the absolute value of w -Yes, is the triangle inequality, where z and w are both complex numbers-
|z+w| $\leqslant$ |z| + |w|

In the proof, the key is that he points out that
$$2Re(z\overline{w}) \leqslant 2|z\overline{w}|$$

which obviously implies that
$$Re(z\overline{w}) \leqslant |z\overline{w}|$$

Why is that so? How does he knows this inequality is satified? If you can help me I would appreciate it a lot.

 

[micromass]

It's from part (d)...

 

[josueortega]

I think I got it now! We know that

$$|Re(x)| \leqslant |x|$$

$$ [Re(x)Re(x)]^{1/2} \leqslant |x|$$

$$ Re(x) \leqslant |x| $$ which is what we wanted to prove. Right?

 

[dylanbyte]

I think that you are slightly overthinking this, the real part of a complex number is just that - a real number.

And the modulus of a real number is just its absolute value, and a real number is always less than its absolute value.

 

[blue_raver22]

Hello,

I can explain the reasoning behind this inequality in Rudin's proof. The key concept here is the concept of "absolute value" or "modulus" of a complex number. The absolute value of a complex number z is defined as the distance from the origin to the point representing z on the complex plane. This distance can be calculated using the Pythagorean theorem, where the real part of z represents the horizontal distance and the imaginary part represents the vertical distance.

Now, in the proof of Theorem 1.33 part e, Rudin is trying to show that the absolute value of z+w is equal or smaller than the sum of the absolute values of z and w. This can be understood geometrically as the distance from the origin to the point representing z+w being smaller or equal to the sum of the distances from the origin to the points representing z and w.

To prove this, Rudin uses the fact that the real part of z+w can be written as the sum of the real parts of z and w, and the imaginary part of z+w can be written as the sum of the imaginary parts of z and w. Using the definition of absolute value, we can write:

|z+w| = √((Re(z) + Re(w))^2 + (Im(z) + Im(w))^2)

Now, using the triangle inequality for real numbers, we can say that:

|z+w| ≤ √((Re(z))^2 + (Im(z))^2) + √((Re(w))^2 + (Im(w))^2)

= |z| + |w|

This is the desired result, and it is satisfied because the real part of z and the real part of w are both smaller than or equal to the absolute values of z and w respectively. This can be seen from the definition of absolute value and the fact that the real part of a complex number is always smaller than or equal to its absolute value.

I hope this explanation helps you understand Rudin's proof better. Let me know if you have any further questions.

Best regards,