A _vector [space](Spaces.md)_, $\mathcal{V}$, is a space defined over a number field $\mathbb{F}$ where we may add or multiply together its elements, such that given any pair of elements $\mathbf{u} \in \mathcal{V}$ and $\mathbf{v} \in \mathcal{V}$ must fulfill the following necessary conditions: 1) $a(\mathbf{u} + \mathbf{v})= a\mathbf{u} + a\mathbf{v}$ and $(a+b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}$ where $a,b \in \mathbb{C}$ or $\in \mathbb{R}$ (closure under addition). ^7c4308 2) $a(b\mathbf{u}) = b(a\mathbf{u})$ (closure under scalar multiplication) ^f3fdb0 3) there exists $\mathbf{0} \in \mathcal{V}$ such that for all $\mathbf{u} \in \mathcal{V}$, $\mathbf{0}\mathbf{u} = 0$ (existence of a 0 element.) ^0ed629 # Dual Spaces With any vector space, we associate a _dual_ vector space, $\mathcal{V}^*$, which contains all [linear map](Linear%20map.md) $\mathbf{u}:\mathcal{V}\rightarrow\mathbb{F}$. ^498b36 %%Ok, cool, but this part needs more info, what sort of map is this? Is this always in inner product map?%% %%Is it possible that dual spaces only exist for inner product spaces?%% # Types of vector spaces * On $\mathbb{R}$ we refer to [real vector spaces](Real%20vector%20spaces.md) * On $\mathbb{C}$ we refer to [complex vector spaces](Complex%20vector%20spaces.md). * [Topological vector spaces](Topological%20vector%20spaces.md) # Vector space structure and construction Here we remark on how vector spaces are constructed. The first step in constructing a vector space is to define a [origin](Geometry%20(index).md#origin) on an [affine space.](Vector%20spaces.md#Affine%20spaces) Specific vector spaces may be then constructed by finding a [basis set](Vector%20spaces.md#Basis%20sets) for that vector space and [vector spaces](Vector%20spaces.md) also contain [vector sub-spaces](Vector%20spaces.md#Vector%20sub-spaces) that are themselves also defined through a basis sets. What this means is that a vector space may be defined as a [direct sum](Vector%20spaces.md#Direct%20Sums%20of%20vector%20spaces) of its subspaces. Or conversely, vector spaces may be formed from direct sums of other vector spaces. %%This allows us to also define hierarchies of vector spaces found in [multi-linear algebra](Tensors%20and%20Multilinear%20Algebra%20(index).md) - what is this saying exactly? Does a hierarchy even mean anything in this context?%% %%Is there any limitation to the direct sum? As in, can you form a new vector space from any direct sum?%% ## Affine spaces Every vector space has an associated [affine space.](Affine%20space.md) ## The zero vector space The _zero vector space_ or _trivial vector space_ is one containing only the [the 0 vector.](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#The%200%20vector) It satisfies the [vector space axioms](Vector%20spaces.md) since $a(\mathbf{0} + \mathbf{0})= a\mathbf{0} + a\mathbf{0}$ ([axiom 1](Vector%20spaces.md#^7c4308)), $a(b\mathbf{0}) = b(a\mathbf{0})$ ([axiom 2](Vector%20spaces.md#^f3fdb0)), and it contains the element, $\mathbf{0}$ ([axiom 3](Vector%20spaces.md#^0ed629)).  ([... see more](Finite%20dimensional%20vector%20spaces.md#0%20dimensional%20vector%20space)) ## Basis sets Every vector space includes at least one [basis.](linear%20basis) %%Axler brings up the notion of a 'standard basis' where a standard basis has vectors that are all 0 except for one element. Does this one basis a vector space must have necessarily also a standard basis? There seems to be no definition for a 'standard basis' https://math.stackexchange.com/questions/4721938/does-every-finite-dimensional-vector-space-have-a-standard-basis%% ### Vector space dimension  ## Subspaces of vector spaces A [subset](Subsets.md) $\mathcal{U}\subseteq \mathcal{V}$ is a [subspace](Spaces.md#Subspaces) if $\mathcal{U}$ is also a vector space. Equivalently, [subset](Subsets.md) of a [vector space](Vector%20spaces.md) is a subspace if it also has [the necessary properties of a vector space.](Vector%20spaces.md) If it is the case that the [subspace](Vector%20spaces.md#Subspaces%20of%20vector%20spaces) $\mathcal{U}$ is also a [proper subset](Subsets.md) ($\mathcal{U}\subset \mathcal{V}$), we refer to $\mathcal{U}$ as a _[proper subspace.](Spaces.md#^f5c603)_ ^e57c5a ## Sums of vector spaces  [(... see more)](Sums%20of%20vector%20spaces.md) ### Direct Sums of vector spaces Consider a [sum of vector spaces,](Vector%20spaces.md#Sums%20of%20vector%20spaces)  [(... see more)](Direct%20sums%20of%20vector%20spaces.md) --- # Proofs and Examples ## Proof that every vector space contains a basis set --- # Recommended reading #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces #MathematicalFoundations/Analysis/FunctionalAnalysis