Functional Analysis

1. Functional Analysis#

Let us introduce some essential definitions. These are taken from [Rudin, 1991], [Evans, 2010], and [Hinze et al., 2008].

Definition 1.1

A vector space (or linear space) over a numeric field \(F\) (such as the real numbers \(\Rbb\) or the complex numbers \(\mathbb{C}\)) is a nonempty set \(V\), whose elements are called vectors, and in which two operations are defined:

  • Addition assigns to vectors \(v \in V\) and \(w \in V\) a third vector, \(v+w \in V\).

  • Multiplication by scalar assigns to a scalar \(a \in F\) and a vector \(v \in V\) another vector, \(a v \in V\).

Furthermore, the following algebraic properties are satisfied:

  1. Commutativity of addition: \(v + w = w + v \quad \forall \, v, w \in V\).

  2. Associativity of addition: \(u + (v + w) = (u + v) + w \quad \forall \, u, v, w \in V\).

  3. Identity vector for the addition: \(\exists \, 0 \in V : \ v + 0 = v \quad \forall \, v \in V\).

  4. Existence of the inverse for the addition: \(\forall \, v \in V \ \exists -v \in V : \ v + (-v) = 0\).

  5. Associativity of scalar multiplication: \(a(bv) = (ab)v \quad \forall \, a, b \in F, \ v \in V\).

  6. Identity element for the scalar multiplication: \(1 v = v \quad \forall \, v \in V\).

  7. Distributivity w.r.t. addition: \(a(v + w) = a v + a w \quad a \in F, \ v, w, \in V\).

  8. Distributivity w.r.t. scalar multiplication: \((a+b)v = a v + b v \quad a,b \in F, \ v \in V\).

Definition 1.2

Let \(V\) be a real vector space. A mapping \(\| \cdot \| : V \to [0, \infty)\) is called a norm if:

  1. \(\| v \| = 0\) if and only if \(v = 0\).

  2. \(\| \lambda v \| = |\lambda| \|v\| \quad \forall \, \lambda \in \Rbb, \ v \in V\).

  3. \(\| v + w \| \le \|v\| + \|w\|\) \(\forall \, v, w \in V\) (triangular inequality).

The tuple \((V, \|\cdot\|)\) is called a normed linear or vector space.

Definition 1.3

Let \(V\) be a normed vector space.

  1. A sequence \(\{v_i\}_{i \in \Nbb} \subset V\) is called a Cauchy sequence if for each \(\varepsilon>0\) there exists \(N>0\) such that

    \[ \| v_i - v_j \| < \varepsilon \qquad \forall i,j > N. \]

  2. A space \(V\) is complete if each Cauchy sequence in \(V\) converges. More precisely, if \(\{v_i\}_{i \in \Nbb}\) is a Cauchy sequence, there exists a \(\bar{v} \in V\) such that \(\{v_i\}_{i \in \Nbb}\) converges to \(\bar{v}\).

  3. A Banach space is a complete, normed vector space.

Definition 1.4

Let \(V\) be a real vector space. A mapping \(( \cdot, \cdot ) : V \times V \to \Rbb\) is called an innter product if:

  1. \((v,w) = (w,v) \quad v, w \in V\).

  2. The mapping \(v \mapsto (v, w)\) is linear for each \(w \in V\).

  3. \((v,v) \ge 0 \quad \forall v \in V\) and the equality only holds for \(v=0\).

Definition 1.5

If \((\cdot, \cdot)\) is an inner product, the associated norm is defined by

\[ \| v \| = (v, v)^{1/2} \qquad v \in V. \]

Definition 1.6

A Hilbert space is a Banach space endowed with an inner product which generates a norm.