List of various vector (linear) spaces

Question

The vector (linear) space is defined as a non-empty set L over a field F, where two relations (binary operations) are defined:

1. Addition

   $ \oplus: L \times L \longrightarrow L $

2. Scalar multiplication

   $ \odot: F \times L \longrightarrow L $

Although we call these relations as addition and scalar multiplication, both of these relations can have arbitrary forms, which do not need to have anything in common with the traditional apprehension of the addition and the multiplication (for example the addition and the multiplication of real numbers). From the set-theoretical point of view are both operations just mappings between two sets, closed under the relations, which need to meet following conditions (axioms of linear space):

1. Associativity of addition
2. Commutativity of addition
3. Identity element of addition
4. Inverse elements of addition
5. Distributivity of scalar multiplication with respect to vector addition
6. Distributivity of scalar multiplication with respect to field addition
7. Compatibility of scalar multiplication with field multiplication
8. Identity element of scalar multiplication

We are usually working with the linear space of complex numbers $\mathbb{C}^n$, because all linear spaces of dimension of $n$ are isomorphic. In other words, we can use linear transformation between two different linear spaces and equivalently solve the problem in some well known linear space (usually $\mathbb{C}^n$) and then transform it back.

My question is: What kinds of linear spaces do you know? What fields and binary operations compose the linear space. If you can, please, also note the physical application of such a linear space.

I will just summarize the ones I know:

1. Vector space $\mathbb{C}^n$, field of complex numbers $\mathbb{C}$

   Let $\mathbf{x},\mathbf{y} \in \mathbb{C}^n$ and $\alpha \in \mathbb{C}$, where $\mathbf{x} = (x_1, x_2, ..., x_n)$ and $\mathbf{y} = (y_1, y_2, ..., y_n)$. Operation of addition $\oplus$ and scalar multiplication $\odot$ are defined as

   $\mathbf{x} \oplus \mathbf{y} \triangleq (x_1 + y_1, x_2 + y_2, ..., x_n + y_n)$

   $\alpha \odot \mathbf{x} \triangleq (\alpha \cdot x_1, \alpha \cdot x_2, ..., \alpha \cdot x_n)$

   (Note: Plus symbol and dot symbol in the brackets denote operations of addition and multiplication of complex numbers.)

2. Function space, field of complex numbers $\mathbb{C}$

   $ f \oplus g \triangleq f(x) + g(x) $

   $ \alpha \odot f \triangleq \alpha f(x)$

3. Vector space of possitive real numbers $\mathbb{R}^+$, field of real numbers $\mathbb{R}$

   $\mathbf{x} \oplus \mathbf{y} \triangleq x \cdot y$

   $\alpha \odot \mathbf{x} \triangleq x^{\alpha}$

4. Vector space of matrices $\mathbb{F}^{m \times n}$ over field $\mathbb{F}$

   $ (\mathbf{X} \oplus \mathbf{Y})_{i,j} \triangleq (\mathbf{X})_{i,j} + (\mathbf{Y})_{i,j}$

   $ (\alpha \odot \mathbf{X})_{i,j} \triangleq \alpha(\mathbf{X})_{i,j} $

Answer 1

Vector spaces appear in many many places. Just to mention a few that are very important in their respective areas:

Every field extension $E:F$ gives rise to $E$ as a vector space over $F$. The dimension of this vector spaces tells us a lot about the size of the extension and its behaviour.

Banach and Hilbert spaces (extremely important structures in modern analysis) are vector spaces with extra structures.

In linear algebra itself, the vector space of all linear transformations from one vector space to another is a very important vector space. The vector space of all $m\times n$ matrices is of course closely related.

The list goes on pretty much forever since vector spaces are extremely important and appear often. An complete list is probably impossible and is certainly pointless. I tried above to just give you a few pointers for very fundamental appearances of vector spaces.