Definition 2.1.1. Let \(k\in\mathbb N\) be positive. If \(x=(x_1,\dots,x_k)\) and \(y=(y_1,\dots,y_k)\) are elements of \(\mathbb R^k,\) and \(\lambda\in\mathbb R,\) we define the coordinatewise operations
\[x+y=(x_1+y_1,\dots,x_k+y_k), \qquad x-y=(x_1-y_1,\dots,x_k-y_k),\]and
\[\lambda x = (\lambda x_1,\dots,\lambda x_k).\]Then, we define the inner product of \(x\) and \(y\) by
\[x\cdot y = \sum_{i=1}^k x_iy_i,\]and the norm of \(x\) by
\(\| x\| = (x\cdot x)^{1/2}.\) We also define the zero tuple of \(\mathbb R^k\) by
\[O=(0,\dots,0).\]Proposition 2.1.2. Let \(x,y,z\in \mathbb R^k\) and \(\lambda \in \mathbb R.\) Then
- \(0\le \|x\|,\)
- \(\|x\|=0 \Leftrightarrow x=\mathit O,\)
- \(\|\lambda x\| = |\lambda|\|x\|.\)
Proof.
Theorem 2.1.3. Let \(x,y,z\in \mathbb R^k\) and \(\lambda \in \mathbb R.\) Then
- \(|x\cdot y|\le \|x\| \|y\|,\)
- \(\|x+y\|\le \|x\|+\|y\|,\)
- \(\|x-z\|\le \|x-y\| + \|y-z\|.\)
References
- Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). McGraw-Hill Education.