Recall that a [[Euclidean spaces|Euclidean space]] is an $\mathbb{R}$-[[Vector spaces|vector space]] $V$ together with a symmetric, positive definite, bilinear form $\tau: V\times V\to \mathbb{R}$ (an *inner product*). The length or Euclidean norm of a vector is then given by function $v\mapsto \sqrt{ \tau(v,v) }$. *Norms* abstract this notion of length. In $\mathbb{R}^{2}$, for example, the we may define the length of vector $(x_{1},x_{2})$ as $\lvert x_{1} \rvert + \lvert x_{2} \rvert$ which is known as the '*taxicab*', '*Manhattan*' or $1$- norm (see [[p-Norms]]). The taxicab norm does not arise from any quadratic form (why? The condition for when a norm arises from quadratic form is known as the [[Parallelogram law|parallelogram law]]). # Definitions ###### Norm axioms Let $X$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$. A norm on $X$ is function $\lVert \cdot \rVert: X\to [0,\infty)$ satisfying the conditions: 1. *Positive definiteness*: for $v\in V$, $\lVert v \rVert = 0 \iff v=0$. 2. *Homogeneity*: for $v\in X$, and $\lambda\in \mathbb{R}$ (or $\mathbb{C}$), $\lVert \lambda v \rVert=\lvert \lambda \rvert \lVert v \rVert$. 3. *Triangle inequality*: for all $v,w\in X$, $\lVert v+w \rVert \leq \lVert v \rVert + \lVert w \rVert$. We say that $(X, \lVert \cdot \rVert)$ is a *normed (vector) space*. Convince yourself that [[Euclidean spaces are normed spaces|these are indeed properties of the Euclidean norm]]. The condition of homogeneity makes it clear why we want our ground field to be $\mathbb{R}$ or $\mathbb{C}$, so that $\lvert \lambda \rvert$ either means absolute value or complex modulus (such a notion may not exist for a field in general). ###### Remarks on the triangle inequality It may not be immediately clear how the triangle inequality is a property of our original Euclidean norm, i.e. $\tau(v+w,v+w)\leq \tau(v,v)+ 2\tau(v,v)\tau(w,w) + \tau(w,w).$Since $\tau(v+w,v+w)=\tau(v,v)+2\tau(v,w)+\tau(w,w)$, it suffices to check that $\tau(v,w)\leq \tau(v,v)\tau(w,w)$ - but this is a standard result from real analysis known as the [[Cauchy-Schwartz inequality]]. Moreover, the triangle inequality is equivalent to [[Reverse triangle inequality|reverse triangle inequality]] which asserts that for all $x,y \in X$, $\lVert x-y \rVert \geq \biggr \lvert \lVert x \rVert - \lVert y \rVert \biggr \rvert .$ The key idea in the proof of the forward direction is that $x=x-y+y$ and $y=y-x+x$ yields $\lVert x \rVert \leq \lVert x -y \rVert + \lVert y \rVert$ and $\lVert y \rVert \leq \lVert y-x \rVert +\lVert x \rVert$. Furthermore, **Homogeneity** yields $\lVert x-y \rVert=\lVert y-x \rVert$ and thus $\lVert x-y \rVert \geq \max \{ \lVert y \rVert - \lVert x \rVert, \lVert x \rVert - \lVert y \rVert \} = \biggr \lvert \lVert x \rVert - \lVert y \rVert \biggr \rvert.$ ###### Convergence We say that a sequence $(x_{n})_{n=1}^{\infty}$ in $X$ [[Convergence|converges]] to $x\in X$ if and only if the sequence $\lVert x_{n} -x \rVert$ of real numbers converges to $0$, as $n\to \infty$. Since $0\leq\biggr\lvert \lVert x_{n} \rVert - \lVert x \rVert \biggr\rvert\leq \lVert x_{n} -x \rVert$, the sandwich rule yields that $\lVert x_{n} \rVert \to \lVert x \rVert$ as $n\to \infty$. ###### The unit ball characterisation of norms Another equivalent condition to $(3)$ is the following (TBC). See [[Equivalence of triangle inequality and convexity of closed unit ball]]. # Properties ###### Equivalence of norms We say that two norms $\lVert \cdot \rVert_{1}$, $\lVert \cdot \rVert_{2}$ on $X$ are *equivalent* if and only if there exist constants $0<c_{1}\leq c_{2}$ such that $c_{1}\lVert x \rVert_{2} \leq \lVert x \rVert_{1} \leq c_{2}\lVert x \rVert_{2}, \quad \forall x\in X.$Equivalently, $\lVert \cdot \rVert_{1}$, $\lVert \cdot \rVert_{2}$ are equivalent if and only if they generate the same open sets. Hence, $x_{n}$ converges to $X$ w.r.t to $\lVert \cdot \rVert_{1}$ if and only if it converges to $x$ w.r.t $\lVert \cdot \rVert_{2}$ (see [[Equivalence of norms]]). (TBC, State equivalence in terms of closed balls). ###### Norms on $\mathbb{R}^{n}$ See [[Equivalence of all norms on real n-space]]. All norms on $\mathbb{R}^{n}$ are equivalent. # Applications We can use normed spaces to develop a notion of [[Continuous maps]]. # Generalisations See [[Metrics]].