Hilbert Space is a mathematical space proposed by David Hilbert, German Mathematician. It is an extension of Euclidean space for infinite dimensions.
Have you ever wondered how physicists understand particles and waves? How do they study them?
Let’s understand with an analogy!
Need for a Mathematical Space
To understand how a ball behaves when it is thrown vertically upwards, it is important to study the surroundings of the ball. This includes factors like acceleration due to gravity, air friction, wind velocity and the like. The surrounding or space where the ball is thrown impacts on the behavior of the ball.
For example, if the ball is thrown on earth, we know that the ball experiences an acceleration due to gravity, of value 
towards the earth. Now, if the same ball were thrown on the surface of the moon, where the force/acceleration due to gravity is reduced, owing to the mass of the moon, we know that the ball would behave differently. Hence, it is very important in physics to understand the nature of the space where the object is being studied. This helps us compute the behavior of that object mathematically, in a convenient manner.
Thus, choosing or defining a particular space makes it easy for a physicist to understand the particle/wave and study their behavior conveniently.
Another analogy similar to the mentioned above would be measuring the weight of an object in the earthly environment, to the extraterrestrial environment. So while studying both the cases, it is very important to note the change in the value of acceleration due gravity, the change in just the value of acceleration it is observed that the value of weight measured changes drastically, despite the fact that the mass remains unchanged. So it is important that when we choose a mathematical space to study an object, it becomes highly important for us to define the space in such a way that the changing parameters are taken into account.
How does a mathematical space work?
Let us assume there is a mathematical space called, Addition Space. Whatever entities I drop into this space, it will add and give us the sum of them all. Say if we have two waves in that space then, the resultant wave would be a simple addition of them.
Two waves being added
In a similar manner, different spaces are defined to study varied particles, objects or waves. Hilbert space is one such mathematical space which makes it convenient for physicists to study quantum particles. Quantum particles possess behavior that cannot be studied under classical laws.
Thus, we can say that particular mathematical space, allows numerous operations based on its nature and dimensions. The universe we live in has curvature and more than one dimension.
Mathematical Spaces and their functionalities
In general, space is a set of points with some relational properties. Space is referred to as a mathematical space since different mathematical operations are performed in them. Some properties of the space are:
- We perform various operations in them, like, adding vectors, intersections, unions, cutting and pasting and measuring different sizes.
- Various relational operations like convergence and proximity relations are performed.
- Transformations are performed in the spaces and many such varied mathematical operations.
History
It all goes back to the study of real numbers, most popularly referred to as Euclidean Space. Initially, during the times of Greeks, the real numbers were considered to be just the integers and rational numbers, until the discovery of 
. This discovery of irrational numbers led to the discovery of transcendental numbers (for example pi and e). This discovery increased the complexity of real numbers, and, we now know it consists majorly of transcendental numbers.
The pun is simpler a concept the more its mathematical intricacy.
We also know how significant the real numbers are, in science today.
Different Types of Spaces
In Euclidean space all geometrical operations could be performed, like measuring distance between two points, studying geometrical shapes and the like. However, our Universe is known to be a non-Euclidean space.
With further discoveries in algebra, geometry, calculus and set theory, a new perception of space for performing operations, began to take shape. Generally, these mathematical spaces broadly studied under four major categories (there are many other mathematical spaces, based on the object/topic of our study).
- Linear Spaces
- Topological Spaces
- Metric Spaces
- Normed and Banach Spaces
Linear Space
The Linear spaces are vector spaces that are spread uniformly in all directions and do not have any curvatures. The derivative of functions at a given point is linear in operation since a derivative at a particular point in a function is nothing but, a tangent.
Visualization of Linear Space (Photo Credit : Quartl/Wikimedia Commons)
Topological Space
The Topological spaces gained momentum in the early 20th century. They provide a general framework for continuity, convergence and compactness of functions. In simple they are used to study small aspect of a larger structure in detail with more clarity. It is famously described as rubber sheet geometry, i.e. study of geometric properties that are insensitive to stretching and shrinking and shrinking without tearing or glueing.
Topological Space
Metric Space
A Metric space is used to measure the distance between two points and general geometric operations. Metric space is a topological space but not all topological spaces are metric spaces.
Banach and Normed Space
The Banach and normed spaces are spaces that incorporate the interaction of both algebra and geometry. Norm generally means the length, thus these spaces provide for more powerful results.
The different mathematical spaces and their mutual association. The smaller circles are part of larger ones, but need not be vice versa! (Photo Credit : Åshild Telle/Wikimedia Commons)
With a glimpse of understanding mathematical spaces, it helps us to understand what a Hilbert space would mean!
Hilbert Space
In physics, we study objects travelling at speeds comparable to that of light under Relativistic mechanics, and other lower and real-world speeds under non-relativistic mechanics. Thus, Hilbert space is useful to understand and study non-relativistic Quantum Mechanics. Hilbert space is a linear space with an operation of the inner product i.e. scalar product and is similar to the metric space in totality. The various wave functions in quantum mechanics, that describe the states of quantum particles live in Hilbert space. It is a fusion of algebra, topology and geometry.
In quantum physics, a particle is considered as an object that is localized in a physical space i.e. 3 Dimensional Euclidean Space. The particle is described in terms of states, observables or expectation values and these are given by vectors in Hilbert Space. Thus, this aids to find the probability density of the quantum particle in the space.
The Euclidean space could accommodate almost all functions but was limited in dimensions i.e. Euclidean space is considered to be a finite dimensioned space. While the Hilbert space is an extension of Euclidean space and an infinite dimensioned space.
Hilbert space is both a vector space and Normed Space (Photo Credit : Åshild Telle/Wikimedia Commons)
Thus, Hilbert space is a mathematical space with infinite dimension. Almost all operations from simple arithmetic to complex mathematical problems can be solved in the Hilbert space. The popular application of Hilbert spaces is in Quantum Mechanics. With the help of vectors, that are commonly referred to as Eigen Vectors, Hilbert spaces help in solving the mysterious behavior of the quantum particles. They allow the quantum particles to undergo transformations by values called Eigenvalues and brings in a relation between these vectors and values to solve a particular quantum problem.
