Hilbert Space- Introduction

Advanced Analysis, Notes 1: Hilbert spaces (basics)
by Orr Shalit
https://noncommutativeanalysis.com/2012/10/20/advanced-analyis-notes-1-hilbert-spaces-basics/




Vector space

Vectors are represented in boldface. scalars in normal font..

A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. In the following, V × V denotes the Cartesian product of V with itself, and → denotes a mapping from one set to another.

The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (The resultant vector is also an element of the set V.)


The second operation, called scalar multiplication · : F × V → V， takes any scalar a and any vector v and gives another vector av. (Similarly, the vector av is an element of the set V. Scalar multiplication is not to be confused with the scalar product, also called inner product or dot product, which is an additional structure present on some specific, but not all vector spaces. Scalar multiplication is a multiplication of a vector by a scalar; the other is a multiplication of two vectors producing a scalar.)

Elements of V are commonly called vectors. Elements of F are commonly called scalars.

In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively.

To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.[1] In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.

Axiom Meaning
Associativity of addition u + (v + w) = (u + v) + w
Commutativity of addition u + v = v + u
Identity element of addition There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of addition For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0.
Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v [nb 2]
Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F.
Distributivity of scalar multiplication with respect to vector addition   a(u + v) = au + av
Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv

https://en.wikipedia.org/wiki/Vector_space