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||}
d) | | z − w | | ≥ | | z | | − | | w | | {\displaystyle ||z-w||\geq ||z||-||w||}
| | z | | = | | ( z − w ) + w | | {\displaystyle \qquad ||z||=||(z-w)+w||}
| | z | | ≤ | | z − w | | + | | w | | {\displaystyle ||z||\leq ||z-w||+||w||}
∴ | | z | | − | | w | | ≤ | | z − w | | {\displaystyle \therefore \qquad ||z||-||w||\leq ||z-w||}
Si z=a+ib, demuestre que R e ( z ) = z + z ¯ 2 y I m ( z ) = z − z ¯ 2 i {\displaystyle Re(z)={\frac {z+{\overline {z}}}{2}}\qquad y\qquad Im(z)={\frac {z-{\overline {z}}}{2i}}}
a) z + z ¯ 2 = a + i b + ( a − i b ) 2 {\displaystyle \qquad {\frac {z+{\overline {z}}}{2}}={\frac {a+ib+(a-ib)}{2}}}
z + z ¯ 2 = 2 a 2 {\displaystyle {\frac {z+{\overline {z}}}{2}}={\frac {2a}{2}}}
∴ z + z ¯ 2 = R e ( a ) {\displaystyle \therefore \qquad {\frac {z+{\overline {z}}}{2}}=Re(a)}
b) z − z ¯ 2 i = a + i b − ( a − i b ) 2 i {\displaystyle \qquad {\frac {z-{\overline {z}}}{2i}}={\frac {a+ib-(a-ib)}{2i}}}
z + z ¯ 2 i = 2 i b 2 i {\displaystyle {\frac {z+{\overline {z}}}{2i}}={\frac {2ib}{2i}}}
∴ z + z ¯ 2 i = I m ( z ) ) {\displaystyle \therefore \qquad {\frac {z+{\overline {z}}}{2i}}=Im(z))}