Basic properties

7. Basic properties#

Definition: A vector space \(V\) over \({\mathbb R}\) is a set with the operations \(+ : V \times V \rightarrow V\) and \(\cdot : {\mathbb R} \times V \rightarrow V\) such that for all \(u,v \in V\) and \(\lambda, \mu \in {\mathbb R}\) there holds

  • \(u + v = v + u\)

  • \((u+v)+w = u + (v+w)\)

  • \(\lambda \cdot (u+v) = \lambda \cdot u + \lambda \cdot v, \quad (\lambda + \mu) \cdot u = \lambda \cdot u + \mu \cdot u\)

Examples are \({\mathbb R}^n\), the continuous functions \(C^0([0,1])\), or the Lebesgue space \(L_2(\Omega)\).

Definition: A normed vector space \((V,\|\cdot\|)\) is a vector space with the operation \(\| . \| : V \rightarrow {\mathbb R}\) being a norm, i.e., for \(u,v \in V\) and \(\lambda \in {\mathbb R}\) there holds

  • \(\| u + v \| \leq \| u \| + \| v \|\)

  • \(\| \lambda \, u \| = | \lambda | \, \| u \|\)

  • \(\| u \| = 0 \Leftrightarrow u = 0\)

Examples are \((C^0, \|\cdot\|_{\sup})\), or \((C^0, \|\cdot\|_{L_2})\).

Definition: In a complete normed vector space, Cauchy sequences \((u_n) \in V^{\mathbb N}\) converge to an \(u \in V\). A complete normed vector space is called Banach space.

Examples of Banach spaces are \((L_2, \|\cdot\|_{L_2})\), \((C^0, \|\cdot\|_{\sup})\), but not \((C^0, \|\cdot\|_{L_2})\).

Definition: The closure of a normed vector-space \((W, \| \cdot \|_V)\), denoted as \(\overline{W}^{\| \cdot \|_V}\) is the smallest complete space containing \(W\).

Example: \(\overline{ C }^{\| \cdot \|_{L_2}} = L_2\).

Definition: A functional or a linear form \(l(\cdot)\) on \(V\) is a linear mapping \(l(\cdot) : V \rightarrow {\mathbb R}\). The canonical norm for linear forms is the dual norm

\[ \| l \|_{V^\ast} := \sup_{0 \neq v \in V} \frac{l(v)}{\|v\|}. \]

A linear form \(l(\cdot)\) is called bounded if the norm is finite. The vector space of all bounded linear forms on \(V\) is called the dual space \(V^\ast\).

An example for a bounded linear form is \(l(\cdot) : L_2 \rightarrow {\mathbb R} : v \rightarrow \int v \, dx\).

Definition: A bilinear form \(A(\cdot,\cdot)\) on \(V \times W\) is a mapping \(A : V \times W \rightarrow {\mathbb R}\) which is linear in \(u\) and in \(v\). It is called symmetric if \(V=W\) and \(A(u,v) = A(v,u)\) for all \(u,v \in V\).

Examples are the bilinear form \(A(u,v) = \int u v \, dx\) on \(L_2\), or \(A(u,v) := u^T A v\) on \({\mathbb R}^n\), where \(A\) is a (symmetric) matrix, or \(b(u,p) = \int \operatorname{div} u \, q \, dx\) with \(V = [C^1(\Omega)]^d\) and \(W=L_2(\Omega)\).

Definition: A symmetric bilinear form \(A(\cdot,\cdot)\) is called an inner product if it satisfies

  • \(A(v,v) \geq 0 \; \forall \, v \in V\)

  • \(A(v,v) = 0 \Leftrightarrow v = 0\)

Often, it is denoted as \((\cdot,\cdot)_A\), \((\cdot,\cdot)_V\), or simply \((\cdot,\cdot)\).

An example on \({\mathbb R}^n\) is \(u^T A v\), where \(A\) is a symmetric and positive definite matrix.

Definition: An inner product space is a vector space \(V\) together with an inner product \((\cdot, \cdot)_V\).

Lemma: Cauchy Schwarz inequality. If \(A(\cdot,\cdot)\) is a symmetric bilinear form such that \(A(v,v) \geq 0\) for all \(v \in V\), then there holds

\[ A(u,v) \leq A(u,u)^{1/2} A(v,v)^{1/2} \]

Proof: For any \(t \in {\mathbb R}\) there holds

\[ 0 \leq A(u-tv, u-tv) = A(u,u) - 2 t A(u,v) + t^2 A(v,v). \]

If \(A(v,v) = 0\), then \(A(u,u)-2tA(u,v) \geq 0\) for all \(t \in {\mathbb R}\), which forces \(A(u,v) = 0\), and the inequality holds trivially. Else, if \(A(v,v) \neq 0\), set \(t = A(u,v) / A(v,v)\), and obtain

\[ 0 \leq A(u,u) - A(u,v)^2 / A(v,v), \]

which is equivalent to the statement. \(\Box\)

Lemma: \(\| v \|_V := (v,v)_V^{1/2}\) defines a norm on the inner product space \((V, (\cdot,\cdot)_V)\).

Definition: An inner product space \((V,(\cdot,\cdot)_V)\) which is complete with respect to \(\|\cdot\|_V\) is called a Hilbert space.

Definition: A closed subspace \(S\) of an Hilbert space \(V\) is a subset which is a vector space, and which is complete with respect to \(\|\cdot \|_V\).

A finite dimensional subspace is always a closed subspace.

Lemma: Let \(T\) be a continuous linear operator from the Hilbert space \(V\) to the Hilbert space \(W\). The kernel of \(T\), \(\operatorname{ker} T := \{ v \in V : T v = 0 \}\) is a closed subspace of \(V\).

Proof: First we observe that \(\operatorname{ker} T\) is a vector space. Now, let \((u_n) \in \operatorname{ker} T^{\mathbb N}\) converge to \(u \in V\). Since \(T\) is continuous, \(T u_n \rightarrow T u\), and thus \(T u = 0\) and \(u \in \operatorname{ker} T\). \(\Box\)

Lemma: Let \(S\) be a subspace (not necessarily closed) of \(V\). Then

\[ S^\bot := \{ v \in V : (v,w) = 0 \; \forall \, w \in S \} \]

is a closed subspace.

The proof is similar to the Lemma on the kernel.

Definition: Let \(V\) and \(W\) be vector spaces. A linear operator \(T : V \rightarrow W\) is a linear mapping from \(V\) to \(W\). The operator is called bounded if its operator-norm

\[ \| T \|_{V \rightarrow W} := \sup_{0 \neq v \in V} \frac{ \| T v \|_W } { \| v \|_V} \]

is finite.

An example is the differential operator on the according space \(\frac{d}{dx} : (C^1(0,1), \| \cdot \|_{sup} + \| \frac{d}{dx} \cdot \|_{sup}) \rightarrow ( C(0,1), \| \cdot \|_{sup})\).

Lemma: A bounded linear operator is continuous.

Proof: Let \(v_n \rightarrow v\), i.e. \(\| v_n - v \|_V \rightarrow 0\).
Then \(\| T v_n - T v \| \leq \| T \|_{V \rightarrow W } \| v_n - v \|_V\) converges to 0, i.e. \(T v_n \rightarrow T v\). Thus \(T\) is continuous.

Definition: A dense subspace \(S\) of \(V\) is such that every element of \(V\) can be approximated by elements of \(S\), i.e.

\[ \forall {\varepsilon > 0} \, \forall { u \in V } \, \exists { v \in S } \mbox{ such that } \| u - v \|_V \leq \varepsilon. \]

Extension principle: Let \(S\) be a dense subspace of the normed space \(V\), and let \(W\) be a complete space. Let \(T : S \rightarrow W\) be a bounded linear operator with respect to the norm \(\| T \|_{V \rightarrow W}\). Then, the operator can be uniquely extended onto \(V\).

Proof: Let \(u \in V\), and let \(v_n\) be a sequence in \(S\) such that \(v_n \rightarrow u\). Thus, \(v_n\) is Cauchy. \(T v_n\) is a well defined sequence in \(W\). Since \(T\) is bounded, \(T v_n\) is also Cauchy. Since \(W\) is complete, there exists a limit \(w\) such that \(T v_n \rightarrow w\). The limit is independent of the sequence, and thus \(T u\) can be defined as the limit \(w\).

Definition: A bounded linear operator \(T : V \rightarrow W\) is called compact if for every bounded sequence \((u_n) \in V^{\mathbb N}\), the sequence \((T u_n)\) contains a convergent sub-sequence.

Lemma: Let \(V, W\) be Hilbert spaces. An operator is compact if and only if there exists a complete orthogonal system \((u_n)\) for \((\operatorname{ker} T)^\bot\) and values \(\lambda_n \rightarrow 0\) such that

\[ (u_n, u_m)_V = \delta_{n,m} \qquad (T u_n, T u_m)_W = \lambda_n \delta_{n,m} \]

This is the eigensystem of the operator \(K : V \rightarrow V^\ast : u \mapsto (T u, T \cdot)_W\).

Proof: (sketch) There exists an maximizing element of \(\frac{ (Tv, Tv)_W } { (v,v)_V }\). Scale it to \(\| v \|_V = 1 \) and call it \(u_1\), and \(\lambda_1 = \frac{ (Tu_1, Tu_1)_W } { (u_1,u_1)_V }\). Repeat the procedure on the \(V\)-complement of \(u_1\) to generate \(u_2\), and so on.