Real Numbers as a Vector Space

Topics: Calculus - Algebra - Vector Space

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(lemma)

Let ${\mathbb{R}}^2=\{(x,y)\mid x,y\in \mathbb{R}\}$ and…

$$+:{\mathbb{R}}^2\times {\mathbb{R}}^2\to {\mathbb{R}}^2$$

$$\cdot :\mathbb{R}\times {\mathbb{R}}^2\to {\mathbb{R}}^2$$

…be defined as…

$$(a,b)+(c,d):=(a+c,b+d)$$

$$s\cdot (a,b):=(sa,sb)$$

Then, ${\mathbb{R}}^2$ is an $\mathbb{R}$ vector space.

In this vector space, we can define things such as a norm.