NORMED VECTOR SPACES AND INNER PRODUCT SPACES

Normed vector spaces and inner product spaces[edit]

Main articles: Normed vector space and Inner product space

"Measuring" vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted $|\mathbf{�}|$ and ${\displaystyle ⟨\mathbf{�},\mathbf{�}⟩,}$ respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm $|\mathbf{�}|:=\sqrt{⟨\mathbf{�},\mathbf{�}⟩}.$ Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.^[39]

Coordinate space ${�}^{�}$ can be equipped with the standard dot product:

$${\displaystyle ⟨\mathbf{�},\mathbf{�}⟩=\mathbf{�}\cdot \mathbf{�}={�}_1{�}_1+\cdots +{�}_{�}{�}_{�}.}$$

In ${\displaystyle {\mathbf{�}}^2,}$ this reflects the common notion of the angle between two vectors $\mathbf{�}$ and ${\displaystyle \mathbf{�},}$ by the law of cosines:

$${\displaystyle \mathbf{�}\cdot \mathbf{�}=\cos \left(\mathrm{\angle }(\mathbf{�},\mathbf{�})\right)\cdot |\mathbf{�}|\cdot |\mathbf{�}|.}$$

Because of this, two vectors satisfying $⟨\mathbf{�},\mathbf{�}⟩=0$ are called orthogonal. An important variant of the standard dot product is used in Minkowski space: ${\displaystyle {\mathbf{�}}^4}$ endowed with the Lorentz product^[40]

$${\displaystyle ⟨\mathbf{�}|\mathbf{�}⟩={�}_1{�}_1+{�}_2{�}_2+{�}_3{�}_3-{�}_4{�}_4.}$$

In contrast to the standard dot product, it is not positive definite: $⟨\mathbf{�}|\mathbf{�}⟩$ also takes negative values, for example, for ${\displaystyle \mathbf{�}=(0,0,0,1).}$ Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of special relativity.

Topological vector spaces[edit]

Main article: Topological vector space

Convergence questions are treated by considering vector spaces $�$ carrying a compatible topology, a structure that allows one to talk about elements being close to each other.^[41][42] Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if $\mathbf{�}$ and $\mathbf{�}$ in $�,$ and $�$ in $�$ vary by a bounded amount, then so do ${\displaystyle \mathbf{�}+\mathbf{�}}$ and ${\displaystyle �\mathbf{�}.}$^{[nb 8]} To make sense of specifying the amount a scalar changes, the field $�$ also has to carry a topology in this context; a common choice are the reals or the complex numbers.

In such topological vector spaces one can consider series of vectors. The infinite sum

$${\displaystyle \sum _{�=1}^{\mathrm{\infty }}{�}_{�}=\underset{�\to \mathrm{\infty }}{lim}{�}_1+\cdots +{�}_{�}}$$

denotes the limit of the corresponding finite partial sums of the sequence ${�}_1,{�}_2,\dots$ of elements of $�.$ For example, the ${�}_{�}$ could be (real or complex) functions belonging to some function space $�,$ in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples.

Unit "spheres" in ${\mathbf{�}}^2$ consist of plane vectors of norm 1. Depicted are the unit spheres in different $�$-norms, for ${\displaystyle �=1,2,}$ and $\mathrm{\infty }.$ The bigger diamond depicts points of 1-norm equal to 2.

A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval ${\displaystyle [0,1],}$ equipped with the topology of uniform convergence is not complete because any continuous function on $[0,1]$ can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem.^[43] In contrast, the space of all continuous functions on $[0,1]$ with the same topology is complete.^[44] A norm gives rise to a topology by defining that a sequence of vectors ${\displaystyle {\mathbf{�}}_{�}}$ converges to $\mathbf{�}$ if and only if

$${\displaystyle \underset{�\to \mathrm{\infty }}{lim}|{\mathbf{�}}_{�}-\mathbf{�}|=0.}$$

Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of functional analysis—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.^[45] The image at the right shows the equivalence of the $1$-norm and $\mathrm{\infty }$-norm on ${\displaystyle {\mathbf{�}}^2:}$ as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.

From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals)  maps between topological vector spaces are required to be continuous.^[46] In particular, the (topological) dual space  consists of continuous functionals  (or to ). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.^[47]