You have to distinguish between mathematical concepts and economic concepts.
Vector space, euclidean space, metric space, are mathematical concepts.
Commodity space is an economic concept.
The most basic spaces are vector spaces and metric spaces.
A vector space on $R$ is defined as a set $V$ (of elements of any nature) such that there are two operations called scalar multiplication and addition, that satisfy a list of certain properties (too long to be described here).
Vector space is the fundamental concept of linear algebra.
The most well-known example of vector space is $R^n$.
A metric space is defined as a set (of elements of any nature) on which is defined a function that is called distance or metric, that satisfies specific axioms (too long to be described here, but you can easily find them on internet).
It is a generalization of the everyday concept of distance.
Vector spaces and metric spaces are two independent concepts: a vector space, on its own, is not a metric space and a metric space, on its own, is not a vector space.
In order to make a vector space a metric space, you have to define on it a distance.
This is the case of an euclidean space, that is a vector space ($R^n$) on which is defined an inner (or scalar) product. It is possible to prove that an inner product induces a distance, so that an Euclidean space is a metric vector space.
An example of metric space that is not a vector space is the so-called discrete space, which is a set (of elements of any nature) on which is introduced a distance called the discrete distance.
The commodity space, instead, is an economic concept, that from a mathematical point of view can be seen as a vector space, in particular $R^n$ if the number of commodities is $n$.
"A commodity vector (or commodity bundle ) is a list of amounts of the different commodities and can be viewed as a point in Rl,the commodity space. We can use commodity vectors to represent an individual's consumption levels.The lth entry of the commodity vector stands for the amount of commodity l consumed”.
That is, the vectors in $R^n$ are interpreted as bundles of goods, but this is an economic interpretation, and not a mathematical concept.
Other properties can be introduced in the concept of commodity bundle, as a relation of total ordering, and others, but these are different from vector space properties.
Now we can answer your previous question:
“By l dimensions ,( of a commodity space) do we mean l directions? “
We saw that, from a mathematical point of view, the commodity space is a vector space, in particular is $R^n$, so we have to relate to the definition of dimension of a vector space.
This definition is a little complicated because implies the concept of base of a vector space (too long to be explained here). In a general vector space, the dimension of the space is defined as the number of vectors that constitutes a base.
In $R^n$ (n-ples of real numbers) one can prove that the dimension is the number of real numbers that forms a vector, that is $n$. So, in a commodity space the dimension is the number of goods that constitute a commodity bundle.