1.4  Euclidean Spaces

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Definition 1.4.1. Let \(k\in\mathbb N^{+}.\) If \(\mathbf x=(x_1,\dots,x_k)\) and \(\mathbf y=(y_1,\dots,y_k)\) are elements of \(\mathbb R^k,\) and \(\lambda\in\mathbb R,\) we define the coordinatewise operations

\[\mathbf x+\mathbf y=(x_1+y_1,\dots,x_k+y_k), \qquad \mathbf x-\mathbf y=(x_1-y_1,\dots,x_k-y_k),\]

and

\[\lambda \mathbf x = (\lambda x_1,\dots,\lambda x_k).\]

Then, we define the inner product of \(\mathbf x\) and \(\mathbf y\) by

\[\mathbf x\cdot \mathbf y = \sum_{i=1}^k x_iy_i,\]

and the norm of \(\mathbf x\) by

\[\| \mathbf x\| = (\mathbf x\cdot \mathbf x)^{1/2}.\]

We also define the zero tuple of \(\mathbb R^k\) by

\[\mathbf 0=(0,\dots,0).\]

Proposition 1.4.2. Let \(x,y,z\in \mathbb R^k\) and \(\lambda \in \mathbb R.\) Then

1. \(0\le \|\mathbf x\|\);
2. \(\|\mathbf x\|=0\) if and only if \mathbf x=\mathbf 0$;
3. \(\|\lambda \mathbf x\| = |\lambda|\|\mathbf x\|\).

Theorem 1.4.3. Let \(\mathbf x,\mathbf y,\mathbf z\in \mathbb R^k\) and \(\lambda \in \mathbb R.\) Then

1. \(|\mathbf x\cdot \mathbf y|\le \|\mathbf x\| \|\mathbf y\|,\)
2. \(\|\mathbf x+\mathbf y\|\le \|\mathbf x\|+\|\mathbf y\|,\)
3. \(\|\mathbf x-\mathbf z\|\le \|\mathbf x-\mathbf y\| + \|\mathbf y-\mathbf z\|.\)

References

1. Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). McGraw-Hill Education.

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