Euclidean spaces

Consider the vector space ${\mathbb{R}}^k$. In addition to the operations and properties already defined for a vector space, we define the following:

• An operation called the inner product of $\mathbf{v}$ and $\mathbf{w}$ such that $\mathbf{v}\cdot \mathbf{w}={\sum }_{i=1}^k{v}_i{w}_i,$ and
• the norm of $\mathbf{x}$ by $|\mathbf{x}|=(\mathbf{x}\cdot \mathbf{x}{)}^{1/2}={\left({\sum }_{i=1}^k{x}_i^2\right)}^{1/2}$.

The structure now defined (the vector space ${\mathbb{R}}^k$ with the inner product and the norm) is called euclidean $k$-space.

Info

A vector space with a norm is called a normed vector space. Thus, you could say a Euclidean space is a normed vector space with an inner product. Not all normed vector spaces are Euclidean spaces. A Euclidean space specifically refers to a vector space equipped with a Euclidean inner product (usually the standard dot product in ${\mathbb{R}}^n$). In contrast, there are normed vector spaces where the norm is not induced by an inner product (e.g., spaces with a non-Euclidean norm like the p-norm for p $\ne$ 2).

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Results

The following follow from the above definition. Suppose $\mathbf{x},\mathbf{y},\mathbf{z}\in {\mathbb{R}}^k$. Then,

• $|\mathbf{x}|\ge 0$
• $|\mathbf{x}|=0$ if and only if $x=0$.
• $|\alpha \mathbf{x}|=|\alpha ||\mathbf{x}|$
• $|\mathbf{x}\cdot \mathbf{y}|\le |\mathbf{x}||\mathbf{y}|$ (The CS inequality)
• $|\mathbf{x}+\mathbf{y}|\le |\mathbf{x}|+|\mathbf{y}|$ (The triangle inequality)