1st major update src/3_vector_spaces.md

src/3_vector_spaces.md

@@ -97,7 +97,7 @@ You might be already familiar with the concept of performing a number of various
 ### Vector products
 
 In addition to multiplying a vector by a scalar, as mentioned above, one can also multiply two vectors among them. 
-There are two types of vector productions, one where the end result is a scalar (so just a number) and the other where the end result   is another vectors. 
+There are two types of vector products; where the end result is a scalar (so just a number) and where the end result is another vector. 
 
 !!! info "Scalar product of vectors" 
     The scalar product of vectors is given by $$ \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .$$
@@ -112,39 +112,41 @@ There are two types of vector productions, one where the end result is a scalar
     Note that this cross-product can only be defined in *three-dimensional vector spaces*. The resulting vector 
     $\vec{c}=\vec{a}\times \vec{b} $ will have as components $c_1 = a_2b_3-a_3b_2$, $c_2= a_3b_1 - a_1b_3$, and $c_3= a_1b_2 - a_2b_1$.
 
-- A special vector is the **unit vector**, which has a norm of 1 *by definition*. A unit vector is often denoted with a hat, rather than an arrow ($\hat{i}$ instead of $\vec{i}$). To find the unit vector in the direction of an arbitrary vector $\vec{v}$, we divide by the norm: 
-$$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$
+### Unit vector and orthogonality
 
-- Two vectors are said to be **orthonormal** of they are perpendicular (orthogonal) *and* both are unit vectors.
+!!! info "Unit vector"
+    A special vector is the **unit vector**, which has a norm of 1 *by definition*. A unit vector is often denoted with a hat, rather than an arrow ($\hat{i}$ instead of $\vec{i}$). To find the unit vector in the direction of an arbitrary vector $\vec{v}$, we divide by the norm: $$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$
 
-Now we are ready to define in a more formal way what are vector spaces,
+!!! info "Orthogonality"    
+    Two vectors are said to be **orthonormal** of they are perpendicular (orthogonal) *and* both are unit vectors.
+
+Now we are ready to define in a more formal way what vector spaces are,
 an essential concept for the description of quantum mechanics.
 
 The main properties of **vector spaces** are the following:
 
-- A vector space is **complete upon vector addition**.
-This property means that if two arbitrary vectors  $\vec{a}$ and $\vec{b}$
-are elements of a given vector space ${\mathcal V}^n$,
-then their addition should also be an element of the same vector space
-  
-$$\vec{a}, \vec{b} \in {\mathcal V}^n, \qquad \vec{c} = (\vec{a} + \vec{b})
-\in {\mathcal V}^n  \, ,\qquad \forall\,\, \vec{a}, \vec{b} \,.$$
-
-- A vector space is **complete upon scalar multiplication**.
-This property means that when I multiply one arbitrary vector  $\vec{a}$,
-element of the vector space ${\mathcal V}^n$,
-by a general scalar $\lambda$, the result is another vector which also belongs
-to the same vector space
-$$\vec{a} \in {\mathcal V}^n, \qquad \vec{c} = \lambda \vec{a}
-\in {\mathcal V}^n \qquad \forall\,\, \vec{a},\lambda \, .$$
-The property that a vector space is complete upon scalar multiplication and vector addition is
-also known as the **closure condition**.
-
-- There exists a **null element** $\vec{0}$ such that $\vec{a}+\vec{0} =\vec{0}+\vec{a}=\vec{a} $.
-
-- **Inverse element**: for each vector $\vec{a} \in \mathcal{V}^n$ there exists another
-element of the same vector space, $-\vec{a}$, such that their addition results
-in the null element, $\vec{a} + ( -\vec{a}) = \vec{0}$. This element it called the **inverse element**.
+!!! ""
+    A vector space is **complete upon vector addition**.
+    This property means that if two arbitrary vectors  $\vec{a}$ and $\vec{b}$
+    are elements of a given vector space ${\mathcal V}^n$,
+    then their addition should also be an element of the same vector space 
+    $$\vec{a}, \vec{b} \in {\mathcal V}^n, \qquad \vec{c} = (\vec{a} + \vec{b}) \in {\mathcal V}^n  \, ,\qquad \forall\,\, \vec{a}, \vec{b} \,.$$
+
+!!! "" 
+    A vector space is **complete upon scalar multiplication**.
+    This property means that when I multiply one arbitrary vector  $\vec{a}$,
+    element of the vector space ${\mathcal V}^n$, by a general scalar $\lambda$, the result is another vector which also belongs to the same vector space $$\vec{a} \in {\mathcal V}^n, \qquad \vec{c} = \lambda \vec{a}
+    \in {\mathcal V}^n \qquad \forall\,\, \vec{a},\lambda \, .$$
+    The property that a vector space is complete upon scalar multiplication and vector addition is
+    also known as the **closure condition**.
+
+!!! ""
+    There exists a **null element** $\vec{0}$ such that $\vec{a}+\vec{0} =\vec{0}+\vec{a}=\vec{a} $.
+
+!!! ""
+    **Inverse element**: for each vector $\vec{a} \in \mathcal{V}^n$ there exists another
+    element of the same vector space, $-\vec{a}$, such that their addition results
+    in the null element, $\vec{a} + ( -\vec{a}) = \vec{0}$. This element it called the **inverse element**.
 
 A vector space comes often equipped with various multiplication operations between vectors, such as the scalar product mentioned above
 (also known as *inner product*), but also  other operations such as the vector product or the tensor product. There are other properties, both for what we are interested in these are sufficient.