It is amusing to recall myself way back in my early years in graduate school mentioning Hilbert space in a similar tune without myself knowing exactly what a Hilbert space is. Now, of course, I know much more than back then, and Hilbert space has become a staple in my research. I now realize how a thorough knowledge in Hilbert space is essential in making progress in foundations of quantum mechanics. It is something that I want my research students to appreciate, and that I wish for everyone.
It is just unfortunate that a rigorous understanding of Hilbert space may not come easily. Fortunately it may not be completely necessary for the starter to come into grips with the rigorous definition of Hilbert space in order to understand quantum mechanics, so we understand Susskind and Friedman’s stance on the Hilbert space. Just an idea, even it may be vague, may already be sufficient for the starter to appreciate what it is. In this post I wish to provide just that idea.
So what is a Hilbert space? Foremost a Hilbert space is built upon a set. And a set is a collection of objects, and the objects themselves are known as elements of the set. However, the set underlying a Hilbert space is not just a hodgepodge of objects but it is a collection whose elements share some common properties that characterize the set itself. For example, a set may consist of real numbers, or an ordered pair of them, or it may consist of functions with specific properties like smoothness.
But more than the common properties of the elements of the set, the set is endowed with two operations involving the sum of two elements of the set, and the product of every element of the set with a set of scalars, such as the real numbers. The operation of addition has the property that the sum of every pair of elements is again an element of the set. Also the operation of multiplication of any vector with a scalar has the property that the product is likewise an element of the set. A set with the said operations of addition and multiplication is called a vector space, and the elements of which are called vectors.
Underneath every Hilbert space is a vector space. But the vector space is not the Hilbert space itself. Additional structures must be endowed to the vector space to turn it into a Hilbert space. These structures follow from some measures on the elements or every pair of elements of a Hilbert space. First is the measure of length of every vector of the vector space, known as the norm. Second is the measure of distance between two vectors, known as the metric. And third is the measure of perpendicularity between two vectors, known as the inner product.
The norm identifies the unique zero vector which is the vector with zero norm. The metric determines whether two vectors are equal or not; when two vectors has a distance, as measured by the metric, equal to zero, they are identified to be the same vector even though they may look different. The inner product determines whether two vectors are perpendicular or not; when the inner product of two vectors is equal to zero, the vectors are identified to be perpendicular or orthogonal.
In a Hilbert space, the metric derives from the norm and the norm derives from the inner product. In short the structure of a Hilbert space is dictated by the inner product. For the underlying vector space to progress into a Hilbert space one has to endow the vector space with an inner product. But a vector space with an inner product is not yet necessarily a Hilbert space. It is called a pre-Hilbert space.
The emergence of Hilbert space from pre-Hilbert space requires the concept of completeness. Consider a sequence of vectors in the given vector space, that is, an ordered collection of infinitely many vectors. If the distance between two consecutive vectors of the sequence gets arbitrarily smaller as we go up the sequence to infinity, the sequence is known as a Cauchy sequence. A pre-Hilbert space is called complete if every Cauchy sequence in it gets arbitrarily near to a unique vector belonging to the given vector space. The unique vector is called the limit of the Cauchy sequence.
Finally, a complete pre-Hilbert space is a Hilbert space. That’s just it.
Now we have an idea of what is a Hilbert space. With that we can outline how one constructs a Hilbert space. One starts by identifying a set and then turning that set into a vector space by defining the sum of every pair of elements of that set and the product of any scalar and an element of that set. This is followed by defining an inner product to the vector space, from which the norm and metric are obtained. This turns the vector space into a pre-Hilbert space. The Hilbert space is now finally constructed by completing the pre-Hilbert space by adjoining all limits of Cauchy sequences to the pre-Hilbert space. The pre-Hilbert space together with the adjoined elements constitutes the Hilbert space.
If that is not enough, you may go for the rigorous definition of the Hilbert space which is found here.