diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml
index aea4bb8..2959957 100755
--- a/config/en/mkdocs.yaml
+++ b/config/en/mkdocs.yaml
@@ -1,4 +1,4 @@
-site_name: My notes
+site_name: Mathematics and physics wiki
 docs_dir: '../../docs/en'
 
 theme:
@@ -110,6 +110,18 @@ nav:
         - 'Compactness': mathematics/functional-analysis/normed-spaces/compactness.md
         - 'Linear operators': mathematics/functional-analysis/normed-spaces/linear-operators.md
         - 'Linear functionals': mathematics/functional-analysis/normed-spaces/linear-functionals.md
+      - 'Inner product spaces': 
+        - 'Inner product spaces': mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md
+        - 'Direct sums': mathematics/functional-analysis/inner-product-spaces/direct-sums.md
+        - 'Orthonormal sets': mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md
+        - 'Total sets': mathematics/functional-analysis/inner-product-spaces/total-sets.md
+        - 'Fourier series':
+          - 'Formalism': mathematics/functional-analysis/inner-product-spaces/fourier-series/formalism.md
+          - 'Convergence': mathematics/functional-analysis/inner-product-spaces/fourier-series/convergence.md
+        - 'Polynomials': 
+          - 'Legendre polynomials': mathematics/functional-analysis/inner-product-spaces/polynomials/legendre-polynomials.md
+          - 'Hermite polynomials': mathematics/functional-analysis/inner-product-spaces/polynomials/hermite-polynomials.md
+          - 'Laguerre polynomials': mathematics/functional-analysis/inner-product-spaces/polynomials/laguerre-polynomials.md
     - 'Topology':
       - 'Fiber bundles': mathematics/topology/fiber-bundles.md
     - 'Calculus': 
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/direct-sums.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/direct-sums.md
new file mode 100644
index 0000000..3b1f3a9
--- /dev/null
+++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/direct-sums.md
@@ -0,0 +1,99 @@
+# Direct sums
+
+> *Definition 1*: in a metric space $(X,d)$, the **distance** $\delta$ from an element $x \in X$ to a nonempty subset $M \subset X$ is defined as
+>
+> $$
+>   \delta = \inf_{\tilde y \in M} d(x,\tilde y).
+> $$
+
+In a normed space $(X, \|\cdot\|)$ this becomes
+
+$$
+    \delta = \inf_{\tilde y \in M} \|x - \tilde y\|.
+$$
+
+> *Definition 2*: let $X$ be a vector space and let $x, y \in X$, the **line segment** $l$ between the vectors $x$ and $y$ is defined as
+>
+> $$
+>   l = \{z \in X \;|\; \exists \alpha \in [0,1]: z = \alpha x + (1 - \alpha) y\}.
+> $$
+
+Using definition 2, we may define the following.
+
+> *Definition 3*: a subset $M \subset X$ of a vector space $X$ is **convex** if for all $x, y \in M$ the line segment between $x$ and $y$ is contained in $M$. 
+
+This definition is true for projections of convex lenses which have been discussed in [optics](). 
+
+We can now provide the main theorem in this section.
+
+> *Theorem 1*: let $X$ be an inner product space and let $M \subset X$ be a complete convex subset of $X$. Then for every $x \in X$ there exists a unique $y \in M$ such that
+>
+> $$
+>   \delta = \inf_{\tilde y \in M} \|x - \tilde y\| = \|x - y\|,
+> $$
+>
+> if $M$ is a complete subspace $Y$ of $X$, then $x - y$ is orthogonal to $X$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Now that the foundation is set, we may introduce direct sums.
+
+> *Definition 4*: a vector space $X$ is a **direct sum** $X = Y \oplus Z$ of two subspaces $Y \subset X$ and $Z \subset X$ of $X$ if each $x \in X$ has a unique representation
+>
+> $$
+>   x = y + z,
+> $$
+>
+> for $y \in Y$ and $z \in Z$. 
+
+Then $Z$ is called an *algebraic complement* of $Y$ in $X$ and vice versa, and $Y$, $Z$ is called a *complementary pair* of subspaces in $X$.
+
+In the case $Z = \{z \in X \;|\; z \perp Y\}$ we have that $Z$ is the *orthogonal complement* or *annihilator* of $Y$. Also denoted as $Y^\perp$. 
+
+> *Proposition 1*: let $Y \subset X$ be any closed subspace of a Hilbert space $X$, then
+>
+> $$
+>   X = Y \oplus Y^\perp,
+> $$
+>
+> with $Y^\perp = \{x\in X \;|\; x \perp Y\}$ the orthogonal complement of $Y$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+We have that $y \in Y$ for $x = y + z$ is called the *orthogonal projection* of $x$ on $Y$. Which defines an operator $P: X \to Y: x \mapsto Px \overset{\mathrm{def}}= y$. 
+
+> *Lemma 1*: let $Y \subset X$ be a subset of a Hilbert space $X$ and let $P: X \to Y$ be the orthogonal projection operator, then we have
+>
+> 1. $P$ is a bounded linear operator,
+> 2. $\|P\| = 1$,
+> 3. $\mathscr{N}(P) = \{x \in X \;|\; Px = 0\}$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Lemma 2*: if $Y$ is a closed subspace of a Hilbert space $X$, then $Y = Y^{\perp \perp}$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Then it follows that $X = Y^\perp \oplus Y^{\perp \perp}$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Lemma 3*: for every non-empty subset $M \subset X$ of a Hilbert space $X$ we have
+>
+> $$
+>   \mathrm{span}(M) \text{ is dense in } X \iff M^\perp = \{0\}.
+> $$
+
+??? note "*Proof*:"
+
+    Will be added later.
\ No newline at end of file
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/convergence.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/convergence.md
new file mode 100644
index 0000000..e69de29
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/formalism.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/fourier-series/formalism.md
new file mode 100644
index 0000000..e69de29
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md
new file mode 100644
index 0000000..c058f84
--- /dev/null
+++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/inner-product-spaces.md
@@ -0,0 +1,122 @@
+# Inner product spaces
+
+> *Definition 1*: a vector space $X$ over a field $F$ is an **inner product space** if an **inner product** $\langle \cdot, \cdot \rangle: X \times X \to F$ is defined on $X$ satisfying
+>
+> 1. $\forall x \in X: \langle x, x \rangle \geq 0$,
+> 2. $\langle x, x \rangle = 0 \iff x = 0$,
+> 3. $\forall x, y \in X: \langle x, y \rangle = \overline{\langle y, x \rangle}$,
+> 4. $\forall x, y \in X, \alpha \in F: \langle \alpha x, y \rangle = \alpha \langle x, y \rangle$,
+> 5. $\forall x, y, z \in X: \langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$.
+
+Similar to the case in normed spaces we have the following proposition.
+
+> *Proposition 1*: an inner product $\langle \cdot, \cdot \rangle$ on a vector space $X$ defines a norm $\|\cdot\|$ on $X$ given by
+>
+> $$
+>   \|x\| = \sqrt{\langle x, x \rangle},
+> $$
+>
+> for all $x \in X$ and is called the **norm induced by the inner product**.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Which makes an inner product space also a normed space as well as a metric space, referring to proposition 1 in normed spaces.
+
+> *Definition 2*: a **Hilbert space** $H$ is a complete inner product space with its metric induced by the inner product.
+
+Definition 2 makes a Hilbert space also a Banach space, using proposition 1.
+
+## Properties of inner product spaces
+
+> *Proposition 2*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space, then
+>
+> $$
+>   \| x + y \|^2 + \| x - y \|^2 = 2\big(\|x\|^2 + \|y\|^2\big), 
+> $$
+>
+> for all $x, y \in X$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Proposition 2 is also called the parallelogram identity.
+
+> *Lemma 1*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space, then
+>
+> 1. $\forall x, y \in X: |\langle x, y \rangle| \leq \|x\| \cdot \|y\|$,
+> 2. $\forall x, y \in X: \|x + y\| \leq \|x\| + \|y\|$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Statement 1 in lemma 1 is known as the Schwarz inequality and statement 2 is known as the triangle inequality and will be used throughout the section of inner product spaces.
+
+> *Lemma 2*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space and let $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ be sequences in $X$, if we have  $x_n \to x$ and $y_n \to y$ as $n \to \infty$, then
+>
+> $$
+>   \lim_{n \to \infty} \langle x_n, y_n \rangle = \langle x, y \rangle.
+> $$
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Completion
+
+> *Definition 3*: an **isomorphism** $T$ of an inner product space $(X, \langle \cdot, \cdot \rangle)_X$ onto an inner product space $(\tilde X, \langle \cdot, \cdot \rangle)_{\tilde X}$ over the same field $F$ is a bijective linear operator $T: X \to \tilde X$ which preserves the inner product
+>
+> $$
+>   \langle Tx, Ty \rangle_{\tilde X} = \langle x, y \rangle_X,
+> $$
+>
+> for all $x, y \in X$. 
+
+As a first application of lemma 2, let us prove the following.
+
+> *Theorem 1*: for every inner product space $(X, \langle \cdot, \cdot \rangle)_X$ there exists a Hilbert space $(\tilde X, \langle \cdot, \cdot \rangle)_{\tilde X}$ that contains a subspace $W$ that satisfies the following conditions
+>
+> 1. $W$ is an inner product space isomorphic with $X$. 
+> 2. $W$ is dense in $X$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Somewhat trivially, we have that a subspace $M$ of an inner product space $X$ is defined to be a vector subspace of $X$ taken with the inner product on $X$ restricted to $M \times M$. 
+
+> *Proposition 3*: let $Y$ be a subspace of a Hilbert space $X$, then
+>
+> 1. $Y$ is complete $\iff$ $Y$ is closed in $X$,
+> 2. if $Y$ is finite-dimensional, then $Y$ is complete,
+> 3. $Y$ is separable if $X$ is separable.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Orthogonality
+
+> *Definition 4*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space, a vector $x \in X$ is **orthogonal** to a vector $y \in X$ if
+>
+> $$
+>   \langle x, y \rangle = 0,
+> $$
+>
+> and we write $x \perp y$. 
+
+Furthermore, we can also say that $x$ and $y$ *are orthogonal*. 
+
+> *Definition 5*: let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space and let $A, B \subset X$ be subspaces of $X$. Then $A$ is **orthogonal** to $B$ if for every $x \in A$ and $y \in B$ we have
+>
+> $$
+>   \langle x, y \rangle = 0,
+> $$
+>
+> and we write $A \perp B$.
+
+Similarly, we may state that $A$ and $B$ *are orthogonal*.
+
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md
new file mode 100644
index 0000000..db11abc
--- /dev/null
+++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/orthonormal-sets.md
@@ -0,0 +1,65 @@
+# Orthonormal sets
+
+> *Definition 1*: an **orthogonal set** $M$ in an inner product space $X$ is a subset $M \subset X$ whose elements are pairwise orthogonal.
+
+Pairwise orthogonality implies that $x, y \in M: x \neq y \implies \langle x, y \rangle = 0$. 
+
+> *Definition 2*: an **orthonormal set** $M$ in an inner product space $X$ is an orthogonal set in $X$ whose elements have norm 1. 
+
+That is for all $x, y \in M$:
+
+$$
+    \langle x, y \rangle = \begin{cases}0 &\text{if } x \neq y, \\ 1 &\text{if } x = y.\end{cases}
+$$
+
+> *Lemma 1*: an orthonormal set is linearly independent.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+In the case that an orthogonal or orthonormal set is countable it can be arranged in a sequence and call it can be called an *orthogonal* or *orthonormal sequence*. 
+
+> *Theorem 1*: let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal sequence in an inner product space $(X, \langle \cdot, \cdot \rangle)$, then
+>
+> $$
+>   \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 \leq \|x\|^2,
+> $$
+>
+> for all $x \in X$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Theorem 1 is known as the Bessel inequality, and we have that $|\langle x, e_n \rangle|$ are called the Fourier coefficients of $x$ with respect to the orthonormal sequence $(e_n)_{n \in \mathbb{N}}$. 
+
+## Orthonormalisation process
+
+Let $(x_n)_{n \in \mathbb{N}}$ be a linearly independent sequence in an inner product space $(X, \langle \cdot, \cdot \rangle)$, then we can use the **Gram-Schmidt process** to determine the corresponding orthonormal sequence $(e_n)_{n \in \mathbb{N}}$. 
+
+Let $e_1 = \frac{1}{\|x_1\|} x_1$ be the first step and let $e_n = \frac{1}{\|v_n\|} v_n$ be the $n$th step with
+
+$$
+    v_n = x_n - \sum_{k=1}^{n-1} \langle x_n, e_k \rangle e_k.
+$$
+
+## Properties
+
+> *Proposition 1*: let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal sequence in a Hilbert space $(X, \langle \cdot, \cdot \rangle)$ and let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence in the field of $X$, then
+>
+> 1. the series $\sum_{n=1}^\infty \alpha_n e_n$ is convergent in $X$ $\iff$ $\sum_{n=1}^\infty | \alpha_n|^2$ is convergent in $X$.
+> 2. if the series $\sum_{n=1}^\infty \alpha_n e_n$ is convergent in $X$ and $s = \sum_{n=1}^\infty \alpha_n e_n$ then $a_n = \langle s, e_n \rangle$.
+> 3. the series $\sum_{n=1}^\infty \alpha_n e_n = \sum_{n=1}^\infty \langle s, e_n \rangle e_n$ is convergent in $X$ for all $x \in X$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Furthermore, we also have that.
+
+> *Proposition 2*: let $M$ be an orthonormal set in an inner product space $(X, \langle \cdot, \cdot \rangle)$, then any $x \in X$ can have at most countably many nonzero Fourier coefficients $\langle x, e_k \rangle$ for $e_k \in M$ over the uncountable index set $k \in I$ of $M$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
\ No newline at end of file
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/hermite-polynomials.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/hermite-polynomials.md
new file mode 100644
index 0000000..e69de29
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/laguerre-polynomials.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/laguerre-polynomials.md
new file mode 100644
index 0000000..e69de29
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/legendre-polynomials.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/polynomials/legendre-polynomials.md
new file mode 100644
index 0000000..e69de29
diff --git a/docs/en/mathematics/functional-analysis/inner-product-spaces/total-sets.md b/docs/en/mathematics/functional-analysis/inner-product-spaces/total-sets.md
new file mode 100644
index 0000000..90e93d3
--- /dev/null
+++ b/docs/en/mathematics/functional-analysis/inner-product-spaces/total-sets.md
@@ -0,0 +1,58 @@
+# Total sets
+
+> *Definition 1*: a **total set** in a normed space $(X, \langle \cdot, \cdot \rangle)$ is a subset $M \subset X$ whose span is dense in $X$. 
+
+Accordingly, an orthonormal set in $X$ which is total in $X$ is called a total orthonormal set in $X$. 
+
+> *Proposition 1*: let $M \subset X$ be a subset of an inner product space $(X, \langle \cdot, \cdot \rangle)$, then
+>
+> 1. if $M$ is total in $X$, then $M^\perp = \{0\}$.
+> 2. if $X$ is complete and $M^\perp = \{0\}$ then $M$ is total in $X$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Total orthornormal sets
+
+> *Theorem 1*: an orthonormal sequence $(e_n)_{n \in \mathbb{N}}$ in a Hilbert space $(X, \langle \cdot, \cdot \rangle)$ is total in $X$ if and only if 
+>
+> $$
+>   \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \|x\|^2,
+> $$
+>
+> for all $x \in X$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Lemma 1*: in every non-empty Hilbert space there exists a total orthonormal set.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Theorem 2*: all total orthonormal sets in a Hilbert space have the same cardinality.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+This cardinality is called the Hilbert dimension or the orthogonal dimension of the Hilbert space.
+
+> *Theorem 3*: let $X$ be a Hilbert space, then
+>
+> 1. if $X$ is separable, every orthonormal set in $X$ is countable.
+> 2. if $X$ contains a countable total orthonormal set, then $X$ is separable.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Theorem 4*: two Hilbert spaces $X$ and $\tilde X$ over the same field are isomorphic if and only if they have the same Hilbert dimension.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
diff --git a/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md b/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md
index a011836..4a0307a 100644
--- a/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md
+++ b/docs/en/mathematics/functional-analysis/normed-spaces/normed-spaces.md
@@ -9,7 +9,7 @@
 
 Also called a *normed vector space* or *normed linear space*. 
 
-> *Definition 2*: a norm on a vector space $X$ defines a metric $d$ on $X$ given by
+> *Proposition 1*: a norm on a vector space $X$ defines a metric $d$ on $X$ given by
 >
 > $$
 >   d(x,y) = \|x - y\|,
@@ -17,9 +17,13 @@ Also called a *normed vector space* or *normed linear space*.
 >
 > for all $x, y \in X$ and is called a **metric induced by the norm**.
 
+??? note "*Proof*:"
+
+    Will be added later.
+
 Furthermore, there is a category of normed spaces with interesting properties which is given in the following definition.
 
-> *Definition 3*: a **Banach space** is a complete normed space with its metric induced by the norm. 
+> *Definition 2*: a **Banach space** is a complete normed space with its metric induced by the norm. 
 
 If we define the norm $\| \cdot \|$ of the Euclidean vector space $\mathbb{R}^n$ by
 
@@ -58,7 +62,7 @@ This adaptation also works for $C$, $l^p$ and $l^\infty$, obviously. Obtaining t
 
 By definition, a subspace $M$ of a normed space $X$ is a subspace of $X$ with its norm induced by the norm on $X$. 
 
-> *Definition 4*: let $M$ be a subspace of a normed space $X$, if $M$ is closed then $M$ is a **closed subspace** of $X$. 
+> *Definition 3*: let $M$ be a subspace of a normed space $X$, if $M$ is closed then $M$ is a **closed subspace** of $X$. 
 
 By definition, a subspace $M$ of a Banach space $X$ is a subspace of $X$ as a normed space. Hence, we do not require $M$ to be complete.
 
@@ -72,7 +76,7 @@ Convergence in normed spaces follows from the definition of convergence in metri
 
 ## Convergent series
 
-> *Definition 5*: let $(x_k)_{k \in \mathbb{N}}$ be a sequence in a normed space $(X, \|\cdot\|)$. We define the sequence of partial sums $(s_n)_{n \in \mathbb{N}}$ by 
+> *Definition 4*: let $(x_k)_{k \in \mathbb{N}}$ be a sequence in a normed space $(X, \|\cdot\|)$. We define the sequence of partial sums $(s_n)_{n \in \mathbb{N}}$ by 
 >
 > $$
 >   s_n = \sum_{k=1}^n x_k,
@@ -108,7 +112,7 @@ From the notion of absolute convergence the following theorem may be posed.
 
 ## Schauder basis
 
-> *Definition 6*: let $(X, \|\cdot\|)$ be a normed space and let $(e_k)_{k \in \mathbb{N}}$ be a sequence of vectors in $X$, such that for every $x \in X$ there exists a unique sequence of scalars $(\alpha_k)_{k \in \mathbb{N}}$ such that
+> *Definition 5*: let $(X, \|\cdot\|)$ be a normed space and let $(e_k)_{k \in \mathbb{N}}$ be a sequence of vectors in $X$, such that for every $x \in X$ there exists a unique sequence of scalars $(\alpha_k)_{k \in \mathbb{N}}$ such that
 >
 > $$
 >   \lim_{n \to \infty} \|x - \sum_{k=1}^n \alpha_k e_k\| = 0,
@@ -165,7 +169,7 @@ As a first application of this lemma, let us prove the following.
 
 In particular, every finite dimensional normed space is complete.
 
-> *Proposition 1*: every finite-dimensional subspace $M$ of a normed space $(X, \|\cdot\|)$ is a closed subspace of $X$. 
+> *Proposition 2*: every finite-dimensional subspace $M$ of a normed space $(X, \|\cdot\|)$ is a closed subspace of $X$. 
 
 ??? note "*Proof*:"
 
@@ -173,7 +177,7 @@ In particular, every finite dimensional normed space is complete.
 
 Another interesting property of finite-dimensional vector space $X$ is that all norms on $X$ lead to the same topology for $X$. That is, the open subsets of $X$ are the same, regardless of the particular choice of a norm on $X$. The details are as follows.
 
-> *Definition 7*: a norm $\|\cdot\|_1$ on a vector space $X$ is **equivalent** to a norm $\|\cdot\|_2$ on $X$ if there exists $a,b>0$ such that
+> *Definition 6*: a norm $\|\cdot\|_1$ on a vector space $X$ is **equivalent** to a norm $\|\cdot\|_2$ on $X$ if there exists $a,b>0$ such that
 >
 > $$
 >   \forall x \in X: a \|x\|_1 \leq \|x\|_2 \leq b \|x\|_1.
@@ -181,7 +185,7 @@ Another interesting property of finite-dimensional vector space $X$ is that all
 
 This concept is motivated by the following proposition.
 
-> *Proposition 2*: equivalent norms on $X$ define the same topology for $X$.
+> *Proposition 3*: equivalent norms on $X$ define the same topology for $X$.
 
 ??? note "*Proof*:"