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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2000 • Question 2

Question

Math 6210 • Real Analysis 2000 • Question 2

Let \(\mathbb C^n=\{z:z=(z_1,\ldots ,z_n)\}\) be the \(n\)-dimensional complex space with norm \(\|z\|=\sum _{k=1}^n |z_k|\). Prove that there is no scalar product on \(\mathbb C^n\) with the property that for each \(z\in \mathbb C^n\), \(\|z\|^2=(z,z)\).

Raw LaTeX

Let \(\mathbb C^n=\{z:z=(z_1,\ldots,z_n)\}\) be the \(n\)-dimensional
complex space with norm \(\|z\|=\sum_{k=1}^n |z_k|\). Prove that there is no
scalar product on \(\mathbb C^n\) with the property that for each \(z\in\mathbb C^n\),
\(\|z\|^2=(z,z)\).

Topic Tags

Hilbert Spaces and Orthogonality