The Hilbert space

Question

I have two questions concerning the Hilbert space:

First: What is an infinite Hilbert space?

Second: What  is the difference between finite and infinite Hilbert subspace?

Answer 1

I guess you mean infinite/finite dimensional?
Every Hilbert space admits an orthonormal basis.
Infinite then just means that this orthonormal basis consists of an infinite number of elements.
Finite, as you can imagine, means that the number of elements is finite.
An example for an infinite dimensional Hilbert space is the sequence space $\ell^2$.

Answer 2

Hilbert Space is a generalization of the familiar Three dimensional Space around us. It simplifies the representation of a quantum state by representing it with a state vector.

Now, just as any 3-d vector can be decomposed into components, we can decompose a given quantum state into components as well. For this, we need to know dimensionality of the Hilbert space. This dimensionality is usually dependent on what physical quantity you want to measure.

For quantities like the spin, we can use a 2-dimensional state vector. This would have two components which correspond to the probability amplitudes for the state to be in the two basis states(Hilbert space basis vectors equivalent to the traditional $\hat{i}$).

On the other hand, for quantities such as position and momenta, there are infinite possibilities. Therefore, they are represented by state vector in an Infinite dimensional Hilbert space.

From Taylor Series expansion, we know that any continuous and differentiable function can be expanded into a series. The infinite dimensional Hilbert space is a generalization of this concept, where the coefficients of Taylor expansions act as components and powers of $x$ as basis vectors. The traditional wave function taught in undergraduate Quantum Mechanics course belongs to this category

Since Normalization is the key for these state vectors, Those state vectors, like those of spin, which can be properly normalized are called proper state vectors while state vectors(read wave function), like those for position, which can only be normalized up to a Dirac Delta function, are called improper state vectors.

For further reading, https://www.amazon.com/Principles-Quantum-Mechanics-R-Shankar/dp/146157675X, Chapter 1 provides a very good and detailed insight into Hilbert Sapces