Banach space is a complete vector space, on which a norm \({\lVert \; \rVert }\) is defined.
- A distance between vectors:
$$d(v,w) =\lVert v-w \rVert$$
- Every Cauchy sequence of vectors always converges to a limit that is also in that space.

Hilbert space is a complete vector space on which …

Let \(V\) be a vector space over R or C with an inner product \({\langle \;, \;\rangle} \). For any vectors \(v\) and \({w \in V}\)
$$ \lVert v \rVert \lVert w \rVert \geq \left| \langle v , w \rangle \right|$$

Proof:
For \(v= 0\) it's obvious that \(\lVert v \rVert \lVert w \rVert = 0 =\left| \langle v , w \rangle \right|\)

Let V be a vector space over \(\mathrm{R}\). Inner product is an operation \({\langle \;, \;\rangle : V \times V \rightarrow \mathrm{R} }\) that satisfies the following properties:

$$\forall_{ v,w\in V}: \langle v, w \rangle =\langle w, v\rangle$$

$$\forall_{ v,w \in V; \alpha \in \mathrm{R}}: \; \langle \alpha v , w \rangle = {\alpha} \langle v , w \rangle $$

A vector space over a field \((S,\cdot,+)\) is a set V with two operations, addition \({\oplus : V \times V \rightarrow V}\) and scalar multiplication \({\odot : S \times V \rightarrow V}\), that satisfy the axioms:

- associativity of addition
$$\forall_{v, w, u \in V}: (v \oplus w) \oplus u = v \oplus ( w \oplus u)$$

- commutativity …

A field is a set S with two operations called addition \({\oplus: S\times S \rightarrow S}\) and multiplication \({\odot: S\times S \rightarrow S}\) which satisfies the following requirements:

- associativity
$$\forall_{a, b, c \in S}: (a \odot b) \odot c = a \odot ( b \odot c)$$
$$\forall_{a, b, c \in S}: (a \oplus b) \oplus c = a \oplus ( b \oplus c)$$

- commutativity …

A group is a set G with an operation \(\odot\) which satisfies four requirements:

- closure
$$\forall_{a, b \in G}: \; a \odot b \in G$$

- associativity
$$\forall_{a, b, c \in G}: (a \odot b) \odot c = a \odot ( b \odot c)$$

- identity element …