Sean w , z ∈ C {\displaystyle w,z\in {C}} Demostrar que:
a) | | z | | = | | z ¯ | | {\displaystyle ||z||=||{\overline {z}}||}
Demostración
Sea z ¯ = a − i b y z = a + i b {\displaystyle {\overline {z}}=a-ib\qquad y\qquad z=a+ib} entonces
| | z ¯ | | = a 2 + ( − b ) 2 y | | z | | = a 2 + b 2 {\displaystyle ||{\overline {z}}||={\sqrt {a^{2}+(-b)^{2}}}\qquad y\qquad ||z||={\sqrt {a^{2}+b^{2}}}}
∴ | | z | | = | | z ¯ | | {\displaystyle \therefore \qquad ||z||=||{\overline {z}}||}
b) | | z w | | = | | z | | | | w | | {\displaystyle \qquad ||zw||=||z||||w||}
| | z w | | 2 = ( z w ) ( z w ¯ ) {\displaystyle ||zw||^{2}=(zw)({\overline {zw}})}
| | z w | | 2 = ( z w ¯ ) ( w z ¯ ) {\displaystyle ||zw||^{2}=(z{\overline {w}})(w{\overline {z}})}
| | z w | | 2 = | | z | | 2 | | w | | 2 {\displaystyle \qquad ||zw||^{2}=||z||^{2}||w||^{2}}
∴ | | z w | | = | | z | | | | w | | {\displaystyle \therefore \qquad ||zw||=||z||||w||}
c) | | z + w | | ≤ | | z | | + | | w | | {\displaystyle \qquad ||z+w||\leq ||z||+||w||}
| | z + w | | 2 = ( z + w ) ( w + z ¯ ) {\displaystyle ||z+w||^{2}=(z+w)({\overline {w+z}})}
| | z + w | | 2 = ( z + w ) ( w ¯ + z ¯ ) {\displaystyle ||z+w||^{2}=(z+w)({\overline {w}}+{\overline {z}})}
| | z + w | | 2 = z z ¯ + z w ¯ ) + w z ¯ + w w ¯ {\displaystyle ||z+w||^{2}=z{\overline {z}}+z{\overline {w}})+w{\overline {z}}+w{\overline {w}}}
| | z + w | | 2 = | | z | | 2 + 2 R e ( z w ) + | | w | | 2 {\displaystyle \qquad ||z+w||^{2}=||z||^{2}+2Re(zw)+||w||^{2}}
| | z + w | | 2 ≤ | | z | | 2 + 2 | | z | | | | w | | + | | w | | 2 {\displaystyle \qquad ||z+w||^{2}\leq ||z||^{2}+2||z||||w||+||w||^{2}}
| | z + w | | 2 ≤ ( | | z | | + | | w | | ) 2 {\displaystyle \qquad ||z+w||^{2}\leq (||z||+||w||)^{2}}
∴ | | z + w | | ≤ | | z | | + | | w | | {\displaystyle \therefore \qquad ||z+w||\leq ||z||+||w||}