Inner product space in the context of Cosine similarity

In mathematics, a Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space, to infinite dimensions. The inner product, which is the analog of the dot product from vector calculus, allows lengths and angles to be defined. Furthermore, completeness means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.

Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space ${\displaystyle V}$ is a basis for ${\displaystyle V}$ whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.

A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers).

A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ${\displaystyle V}$ with finite dimension is a basis for ${\displaystyle V}$ whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for ${\displaystyle \mathbb {R} ^{n}}$ arises in this fashion.An orthonormal basis can be derived from an orthogonal basis via normalization.The choice of an origin and an orthonormal basis forms a coordinate frame known as an orthonormal frame.

For a general inner product space ${\displaystyle V,}$ an orthonormal basis can be used to define normalized orthogonal coordinates on ${\displaystyle V.}$ Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of ${\displaystyle \mathbb {R} ^{n}}$ under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.

In mathematics, a normed vector space or normed space is a vector space, typically over the real or complex numbers, on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If ${\displaystyle V}$ is a vector space over ${\displaystyle K}$, where ${\displaystyle K}$ is a field equal to ${\displaystyle \mathbb {R} }$ or to ${\displaystyle \mathbb {C} }$, then a norm on ${\displaystyle V}$ is a map ${\displaystyle V\to \mathbb {R} }$, typically denoted by ${\displaystyle \lVert \cdot \rVert }$, satisfying the following four axioms:

1. Non-negativity: for every ${\displaystyle x\in V}$,${\displaystyle \;\lVert x\rVert \geq 0}$.
2. Positive definiteness: for every ${\displaystyle x\in V}$, ${\displaystyle \;\lVert x\rVert =0}$ if and only if ${\displaystyle x}$ is the zero vector.
3. Absolute homogeneity: for every ${\displaystyle \lambda \in K}$ and ${\displaystyle x\in V}$,$${\displaystyle \lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert }$$
4. Triangle inequality: for every ${\displaystyle x\in V}$ and ${\displaystyle y\in V}$,$${\displaystyle \|x+y\|\leq \|x\|+\|y\|.}$$

If ${\displaystyle V}$ is a real or complex vector space as above, and ${\displaystyle \lVert \cdot \rVert }$ is a norm on ${\displaystyle V}$, then the ordered pair ${\displaystyle (V,\lVert \cdot \rVert )}$ is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by ${\displaystyle V}$.

In quantum physics, unitarity is (or a unitary process has) the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space.