Complex Conjugation is not Linear Mapping
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Theorem
Let $\overline \cdot: \C \to \C: z \mapsto \overline z$ be the complex conjugation over the field of complex numbers.
Then complex conjugation is not a linear mapping.
Proof
\(\ds \overline {i \cdot 1}\) | \(=\) | \(\ds \overline i \cdot \overline 1\) | Product of Complex Conjugates | |||||||||||
\(\ds \) | \(=\) | \(\ds - i \cdot \overline 1\) | Definition of Complex Conjugate | |||||||||||
\(\ds \) | \(\ne\) | \(\ds i \cdot \overline 1\) |
By definition, it is not a linear mapping.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations