diff --git a/src/3_vector_spaces.md b/src/3_vector_spaces.md
index 63a62b495309163d381d6c21ddc7062205f491a6..0b86d444cda6c1fcd9062f14d5afd6509c60e5b5 100644
--- a/src/3_vector_spaces.md
+++ b/src/3_vector_spaces.md
@@ -82,34 +82,34 @@ solution proces.
 You might be already familiar with the concept of performing a number of various **operations** between vectors, so in this course, let us review some essential operations that are relevant to start working with quantum mechanics:
 
 !!! info "Addition" 
-  I can add two vectors to produce a third vector, $$\vec{a} + \vec{b}= \vec{c}$$.
-  As with scalar addition, also vectors satisfy the commutative property, $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$.
-  Vector addition can be carried out in terms of their components,
-  $$ \vec{c} = \vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n) =  (c_1, c_2, \ldots, c_n) \, .$$
+    I can add two vectors to produce a third vector, $$\vec{a} + \vec{b}= \vec{c}$$.
+    As with scalar addition, also vectors satisfy the commutative property, $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$.
+    Vector addition can be carried out in terms of their components,
+    $$ \vec{c} = \vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n) =  (c_1, c_2, \ldots, c_n) \, .$$
 
 !!! info "Scalar multiplication" 
-  I can multiply a vector by a scalar number (either real or complex) to produce another vector, $$\vec{c} = \lambda \vec{a}$$. 
-  Addition and scalar multiplication of vectors are both *associative* and *distributive*, so the following relations hold
-  1. $(\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$
-  2. $\lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$
-  3. $(\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$
+    I can multiply a vector by a scalar number (either real or complex) to produce another vector, $$\vec{c} = \lambda \vec{a}$$. 
+    Addition and scalar multiplication of vectors are both *associative* and *distributive*, so the following relations hold
+    1. $(\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$
+    2. $\lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$
+    3. $(\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$
 
 !!! info "Vector product" 
-  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. 
+    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. 
 
 !!! info "The scalar production of vectors" 
-  The **scalar production of vectors is given by $$ \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .$$
-  Note that since the scalar product is just a number, its value will not depend on the specific
-  basis in which we express the vectors: the scalar product is said to be *basis-independent*. The scalar product is also found via 
-  $$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \theta$$ with $\theta$ the angle between the vectors.
+    The **scalar production of vectors is given by $$ \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .$$
+    Note that since the scalar product is just a number, its value will not depend on the specific
+    basis in which we express the vectors: the scalar product is said to be *basis-independent*. The scalar product is also found via 
+    $$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \theta$$ with $\theta$ the angle between the vectors.
 
 !!! info "Cross product"
-  "The vector product (or cross product) between two vectors $\vec{a}$ and $\vec{b}$ is given by 
-  $$ \vec{a}\times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$$
-  where $|\vec{a}|=\sqrt{ \vec{a}\cdot\vec{a} }$ (and likewise for $|\vec{b}|$) is the norm of the vector $\vec{a}$, $\theta$ is the angle between the two vectors, and $\hat{n}$ is a unit vector which is *perpendicular* to the plane that contains $\vec{a}$ and $\vec{b}$. 
-  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 $c_3= a_1b_2 - a_2b_1$.
+    The vector product (or cross product) between two vectors $\vec{a}$ and $\vec{b}$ is given by 
+    $$ \vec{a}\times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$$
+    where $|\vec{a}|=\sqrt{ \vec{a}\cdot\vec{a} }$ (and likewise for $|\vec{b}|$) is the norm of the vector $\vec{a}$, $\theta$ is the angle between the two vectors, and $\hat{n}$ is a unit vector which is *perpendicular* to the plane that contains $\vec{a}$ and $\vec{b}$. 
+    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: 
 $$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$