3.2.1. Distance

The notion of distance allows the definition of neighbors and quantifies as far or as close are two elements.

DEFINITION 3.3.– Let d: V × V → be a function such that

Then, d is a distance on V.

As previously observed, we are not interested in the numeric value of a distance, but in the definition of families of neighbors in order to approximate solutions and study convergence of sequences. For such a purpose, the numeric value of a distance is not useful as information. For instance, if d is a distance on V and η: → is a strictly increasing application such that η(0) = 0 and η(a + b) ≤ η(a) + η(b), then d_η,(u, v) = η(d(u, v)) is a distance on V (see exercises). Thus, we may give an arbitrary value to the distance between two elements of V. Moreover, distances may be pathological: for instance, the defination

corresponds to a distance. This distance is not useful, since the small neigborhoods of u contain a single element, which is u itself: d(u, v)≤ ε ≤1 ⇒ v = u. Thus, V(u; ε) = {u} and the associated topology contains just a few converging sequences and neighborhoods are poor in elements.

Distances allow the definition of limits and continuity. Since our presentation focuses on Hilbert spaces, these notions are introduced later, when considering norms.

3.2.2. Norm

The notion of norm allows the definition of the length of a vector and, at the same time, may be used in order to define a distance. In Mechanics, norms may also be interpreted in terms of internal energy of a system.

DEFINITION 3.4.– Let : V → be a function such that

Then, is a norm on V.

Classical examples of norms on V = ⁿ are (x = (x₁, …, x„)):

THEOREM 3.2.– Let be a norm on V. Then d(u, v) = u − v is a distance on V.

PROOF.– Immediate.

When considering functional spaces, such as subspaces of V = {v: Ω ⊂ ⁿ → ^k} continuity may be an essential assumption in order to define norms. For instance, let us consider p ∈ * and

Let S = {v ∈ V: v is continuous on Ω and v < ∞ }, then is a norm on S. The assumption of continuity is used in order to show that

Indeed, for v continuous on Ω:

If x ∈ Ω is such that v(x)^p > 0, the, by continuity, there is ε > 0 such that v(y)^p>0 on B_ε(x) = {y ∈ Ω: y − x ≤ ε} (Corollary 3.1). Then,

and we have 0 > 0, what is a contradiction. The classical counterexample establishing that continuity must be assumed is furnished by a function v such that v(x) = 0 everywhere on Ω, except on a finite or enumerable number of points. For instance,

Here, v = 0, but v≠ 0. In the following, we show how to we lift this restriction by using equivalence relations that generate the functional space L^p(Ω) − there will be a price to pay: it will be the loss of the concept of value at a point: the individual value v (x) of v in a point x has no meaning for elements of L^p(Ω).

As an alternative norm, we may also consider

In this case, the assumption of continuity is not necessary. Indeed,

3.2.3. Scalar product

The notion of scalar product allows the definition of angles between vectors and, at same time, may be used in order to define a norm. In Mechanics, scalar products may also be interpreted as the virtual work of internal efforts.

DEFINITION 3.5.– Let (, ): V × V → be a function such that

Then, (, ) is a scalar product on V.

A classical example of scalar product on V = ⁿ is (x = (x₁, …, x_n), y = (y₁, …, y_n))

THEOREM 3.3.– Let (, ): V × V → be a scalar product on V. Then, ∀u, v, ∈ V, λ ∈ ,

1. i)
2. ii)
3. iii)
4. iv)
5. v)
6. vi)

PROOF— We have

so that,

and we have (i). Thus:

By adding these two equalities, we have (ii).

Let f(λ) = (u + λv, u + λv). We have f(λ) ≥ 0, ∀λ ≥ 0. Thus,

and we have (iii).

Since

We have

what implies (iv). Replacing v by − v, we obtain (v).

Finally, we observe that

so that (from(iii))

and we have (vi).

THEOREM 3.4.— Let (, ): V × V → be a scalar product on V. Then, is a norm on V.

PROOF — we have

Finally, 3.2.5. (v) implies that u + v ≤ u + v.

Functional spaces, such as subspaces of generally involve continuity assumptions in order to define scalar products. Analogously to norms, the assumption of continuity is necessary in order to verify that

For instance, if

and S = {v ∈ V: v is continuous on Ω and (y, v) < ∞}, then (, ) defines a scalar product on S. Analagously to the case of norms (use p = 2),

In the following, continuity assumption is lifted by using functional space L²(Ω). As previously observed, the consequence is the loss of the meaning of point values v(x)

This limitation may be avoided in particular situations where the scalar product implies the continuity. For instance, let us consider,

where A: B = Σ_i, _j A_ij B_ij. In this case, the continuity of the elements u, v yields from , the existence of the gradients ∇u, ∇v and we may give a meaning to the punctual values u(x) and (x) .

DEFINITION 3.6.— The angle θ(u, v) between the vectors u and v is

u and v are orthogonal if and only if (u, v) = 0 Notation: u ⊥v. For S ⊂ V, we denote by S^⊥ the set of the elements of V which are orthogonal to all the elements of S:

DEFINITION 3.7.– Let F = {φ_n : n ∈ Λ ⊂ ^*} be a countable family. We say that F is an orthogonal family if and only if

We say that F is an orthonormed family if and only if it is orthogonal and, moreover,

A countable free family G = {ø_n: n ∈ Λ ⊂ ^*} may be transformed into an orthonomal family F = {φ_n: n ∈ Λ ⊂ ^*} by the Gram–Schmidt procedure, which has been created for the solution of least-square linear problems. It has been proposed by Jorgen Pedersen Gram [GRA 83] and formalized by Ehrlich Schmidt [SCM 07] (although early uses by Pierre Simon de Laplace have been noticed [LEO 13]). The Gram–Schmidt procedure reads as:

1. 1) Set
   
2. 2) For k > 0, set
   

In this case,

The practical implementation requests the evaluation of scalar products and norms.

As an example, let us consider Ω = (−1, 1) and

The family ø_n(x) = xⁿ is not orthonormal, since (mod(p, q) denotes the remainder of the integer division p/q)

If the Gram–Schmidt procedure is applied, we have:

3.2.4. Cartesian products of vector spaces

In practice, we often face situations where V = V₁ × … V_d is the Cartesian product of vector spaces. In this case, elements u, are given by:

In such a situation:

• – If the distance on V_i is d_i, then the distance d„ on V may be defined as:
  
• – Analogously, if the norm on V_t is , then the norm _v on V may be dfined as:
  
• – Finally, if the scalar product on V_i is (, )_i, then the scalar product (, )_v on V may be defined as:
  

  When , we have and

  

3.2.5. A Matlab® class for scalar products and norms

Below we give an example of a Matlab® class for the evaluation of scalar products and norms. The class contains two scalar products:

Evaluated by methods sp0 and sp1, respectively. The associated norms are evaluated by methods n0 and n1, respectively. For each method, the functions are furnished by structured data. In both the cases, the structure has as properties dim, dimx and component. dim and dimx are integer values defining the dimension of the function and the dimension of its argument x. component is a cell array of length dim containing the values of the components and their gradients: each element of component is a structure having properties value and grad. When using ‘subprogram’, value and grad are Matlab subprograms (i.e. anonymous functions). For ‘table’, value is an array of values and grad is a cell arrays containing tables of values: u.component{i}.value is a table of values of component u_i; and u. component {i} .grad {j} is a table of values of ∂u_i/∂x_j.

For ‘subprogram’, parameter xl contains the limits of the region, analogous to those used in class subprogram_integration. For ‘table’, xl is a structure containing the points of integration, analogously to those used in class riemann.

EXAMPLE 3.2.– Let us evaluate both the scalar products and norms for the functions u, v: (0, 1) → given by u(x) = x, v(x) = x². The functions may be defined as follows:

The code:

xlim.lower.x = 0;
xlim.upper.x = 1;
xlim.dim = 1;
u = u1();
v = v1();
sp = scalar_product();
sp0uv = sp.sp0(u,v, xlim,‘subprogram’);
n0u = sp.n0(u,xlim,‘subprogram’);
n0v = sp.n0(v,xlim,‘subprogram’);
sp1uv = sp.sp1(u,v, xlim,‘subprogram’);
n1u = sp.n1(u,xlim,‘subprogram’);
n1v = sp.n1(v,xlim,‘subprogram’);

produces the results sp0uv = 0.25, n0u = 0.57735, n0v = 0.44721, sp1uv = 1.25, n1u = 1.1547, n1v = 1.2383.

The exact values are, respectively,

We generate cell arrays containing tables of values by using:

x = 0:0.01:1;
p.x = x;
p.dim = 1;
U.dim = 1;
U.dimx = 1;
U.component = cell(u.dim,1);
U.component{1}.value =
spam.partition(u.component{1}.value,p);
U.component{1}.grad = cell(u.dim,1);
U.component{1}.grad{1} =
spam.partition(u.component{1}.grad,p);
V.dim = 1;
V.dimx = 1;
V.component = cell(v.dim,1);
V.component{1}.value =
spam.partition(v.component{1}.value,p);
V.component{1}.grad = cell(v.dim,1);
V.component{1}.grad{1} =
spam.partition(v.component{1}.grad,p);

Then, the code:

sp0UV = sp.sp0(U,V,p,'table');
n0U = sp.n0(U,p,'table');
n0V = sp.n0(V,p,'table');
sp1UV = sp.sp1(U,V,p,'table');
n1U = sp.n1(U,p,'table');
n1V = sp.n1(V,p,'table');

produces the results sp0UV = 0.25003, n0U = 0.57736, n0V = 0.44725, sp1UV = 1.25, n1U = 1.1547, n1V = 1.2383.

EXAMPLE 3.3.— Let us evaluate both the scalar products and norms for the functions u, v: (0, 1) × (0, 2) → given by . The functions may be defined as follows:

The code:

xlim.lower.x = 0;
xlim.upper.x = 1;
xlim.lower.y = 0;
xlim.upper.y = 2;
xlim.dim = 2;
u = u2();
v = v2();
sp = scalar_product();
sp0uv = sp.sp0(u,v, xlim,'subprogram');
n0u = sp.n0(u,xlim,'subprogram');
n0v = sp.n0(v,xlim,'subprogram');
sp1uv = sp.sp1(u,v, xlim,'subprogram');
n1u = sp.n1(u,xlim,'subprogram');
n1v = sp.n1(v,xlim,'subprogram');

produces the results sp0uv = 1.8333, n0u = 0.94281, n0v = 2.0976, sp1uv = 4.8333, n1u = 2.0548, n1v = 3.0111.

The exact values are, respectively

We use the function

function v = select_index(ind,u,x)
aux = u(x);
v = aux(ind);
return;
end

and we generate cell arrays containing tables of values by the code below:

x = 0:0.01:1;
y = 0:0.01:2;
p.x = x;
p.y = y;
p.dim = 2;
U.dim = 1;
U.dimx = 2;
U.component = cell(u.dim,1);
U.component{1}.value =
spam.partition(u.component{1}.value,p);
U.component{1}.grad = cell(u.dim,1);
V.dim = 1;
V.dimx = 2;
V.component = cell(u.dim,1);
V.component{1}.value =
spam.partition(v.component{1}.value,p);
V.component{1}.grad = cell(u.dim,1);
for i = 1: u.dim
  for j = 1: p.dim
    du = @(x) select_index(j,u.component{i}.grad,x);
    U.component{i}.grad{j} = spam.partition(du,p);
    dv = @(x) select_index(j,v.component{i}.grad,x);
    V.component{i}.grad{j} = spam.partition(dv,p);
  end;
end
sp0UV = sp.sp0(U,V,p,'table');
n0U = sp.n0(U,p,'table');
n0V = sp.n0(V,p,'table');
sp1UV = sp.sp1(U,V,p,'table');
n1U = sp.n1(U,p,'table');
n1V = sp.n1(V,p,'table');

It produces the results sp0UV = 1.8334, n0U = 0.94284, n0V = 2.0977, sp1UV = 4.8334, n1U = 2.0548, n1V = 3.0111.

3.2.6. A Matlab® class for Gram–Schmidt orthonormalization

The class defined in the preceding section may be used in order to transform a general family G = {ø_i: 1 ≤ i ≤ n} into an orthonomal family F = {φ_i: 1 ≤ i ≤ n} by the Gram–Schmidt procedure.

Let us assume that the family G is defined by a cell array basis such that basis{i} defines ø_i, according to the standard defined in section 3.2.5 : basis{i} is a structure having properties dim, dimx and component. component is a structure having fields value and grad. The class given in program 3.4 generates a cell array hilbert_basis defining the family F, resulting from the application _ the Gram–Schmidt’s procedure to G. The class uses subprograms sp_space and n_space, which are assumed to evaluate the scala product and the norm.

EXAMPLE 3.4.— Let us consider the family ø_i(x) = xⁱ⁻¹

The family is created by the code

basis = cell(n,1);
for i = 1: n
  basis{i} = u3(i);
end;

We apply the procedure on (−1, 1) with the scalar product sp0:

xlim.lower.x = -1;
xlim.upper.x = 1;
xlim.dim = 1;
sp_space = @(u,v)
scalar_product.sp0(u,v,xlim,'subprogram');
n_space = @(u)
scalar_product.n0(u,xlim,'subprogram');
hb =
gram_schmidt.subprograms(basis,sp_space,n_space);

The results are shown in Figure 3.1 below:

We generate discrete data associated with these functions by using the code:

x = -1:0.02:1;
p.x = x;
p.dim = 1;
tb = cell(n,1);
for i = 1: n
  u = u3(i);
  U.dim = 1;
U.dimx = 1;
  U.component = cell(u.dim,1);
  U.component{1}.value =
spam.partition(u.component{1}.value,p);
  U.component{1}.grad = cell(u.dim,1);
  U.component{1}.grad{1} =
spam.partition(u.component{1}.grad,p);
  tb{i} = U;
end;

Then, the commands

sp_space = @(u,v)
scalar_product.sp0(u,v,p,'table');
n_space = @(u) scalar_product.n0(u,p,'table');
hb = gram_schmidt.tables(tb,sp_space,n_space);

furnish the result shown in Figure 3.2 below:

When the procedure is applied with the scalar product sp1, the results differ. For instance, we obtain the results exhibited in Figures 3.3 and 3.4:

As an alternative, we may define the basis by using the classes previously introduced (section 2.2.6). For instance,

zero = 1e-3;
xmin = -1;
xmax = 1;
degree = 3;
bp = polynomial_basis(zero,xmin,xmax,degree);
%
basis = cell(n,1);
for i = 1: n
  basis{i}.dim = 1;
  basis{i}.dimx = 1;
  basis{i}.component = cell(1,1);
  basis{i}.component{1}.value = @(x) bp.value(i,x);
  basis{i}.component{1}.grad = @(x) bp.d1(i,x);
end;

This definition leads to the same results.

3.4.1. Sequences

DEFINITION 3.9.— A sequence of elements of V (or a sequence on V) is an application ^*∋n → u_n ∈ V. Such a sequence is denoted {u_n: n ∈ W} ⊂ V .

Let * ∋ k → n_k ∈ * be a strictly increasing map. Then * ∋ ∈ V is a subsequence ⊂ {u_n: n ∈^*}

We observe that n_k → +∞ when k → +∞ : we have n_k+1 > n_k so that n_k+1 ≥n_k + l. Thus, n_k+1 ≥ n₁+ k≥k and n_k → +∞ .

In the context of engineering, the main interest of sequences the main interest lies in their use in the construction of approximations, whether for functions (for instance, signals) or solutions of equations. Approximations are connected to the notion of convergence. In the following, we present this concept and we distinguish strong and weak convergence.

3.4.2. Convergence (or strong convergence)

DEFINITION 3.10.– Let V be a vector space having norm . {u_n: n ∈ ^*} ⊂ V converges (or strongly converges) to u ∈ V if and only if u_n − u → 0, when n → ∞. Notation: u_n → u or u = lim_{n→ +∞} u_n.

EXAMPLE 3.5.— Let us consider u_n: (0,1) → given by

and

We have

Thus, u_n → u on V = {v: Ω → : v is continuous on Ω, and v_p < ∞ }. Notice that the limit u ∉ V, since it is discontinuous at x = 1/2, while u_n ∈ V is continuous on Ω, ∀n.

THEOREM 3.5.− u_n → u if and only if , for any subsequence.

PROOF.– Assume that u_n − u → 0. Then:

Let k₀(ε) be such that . Then

Thus, . However, assume that , for any subsequence. Let us suppose that u_n u. Then, there exists ε > 0 such that,

Let us generate a subsequence as follows: n₁ = n(1) and n_k+1 = n(n_k), ∀ k ≥ 1. Then

Thus, 0 < ε ≤ 0, what is a contradiction. So, u_n → u.

We have also

THEOREM 3.6.– LetV, W be two vector spaces having norms _v, _w, respectively. F: V → W is continuous at u ∈ V if and only if u_n → u ⇒ F(u_n) → F(u) .

PROOF.– See [SOU 10].

One of the main differences between vector spaces formed by functions (i.e. functional spaces) and standard finite dimensional spaces such as ⁿ is the fact that bounded sequences may do not have convengent subsequences (see examples 3.6 and 3.7 below).

EXAMPLE 3.6.– Let us consider Ω = (0, 2π),

and S = {v ∈ V: v is continuous on Ω and (v, v) < ∞}. Let us consider the sequence defined by

Let v₀ = 1. We have

Let v_k(x) = x^k (k ≥ 1). We have

Thus, ∀k ≥ 1,

Thus, (u_n, v_k) → 0, ∀k ≥ 0, so that

(u_n, P) → 0, ∀ polynomial function P: (0, 2n) → .

Let v: (0, 2π) → be a continuous function. From Stone–Weierstrass approximation theorem (see theorem 3.13), there is a polynomial P_ε: (0, 2π) → such that v − P_ε∞ ≤ ε. Considering that

we have

and

Since ε > 0 is arbitrary, we have (u_n, v) → 0. Assume that . is a subsequence of {(u_n, v): n ∈ ^*}, so that . We also have

so that . Thus, (u, v) = 0, ∀v, ∈ S. Taking v = u, we have (u, u) = 0. This equality implies that u = 0. But

so that and we have , which is a contradiction. We conclude that the sequence {u_n: n ∈ ^*} has no converging subsequence.

3.4.3. Weak convergence

DEFINITION 3.11.– Let V be a vector space having scalar product (, ). {u_n: n∈^*} ⊂ V weakly converges to u ∈ V if and only if ∀v ∈ V: (u_n − u, v) → 0, when n → +∞. Notation: u_n u.

EXAMPLE 3.7.– Let us consider Ω = (0, 1)

and u_n: (0, 1) → given by u_n(x) = cos (2nπx). Let v₀ = 1. We have

Let v_k(x) = x^k (k≥1). We have:

So that

Thus,

Let v be continuous on [0, 1] and ε > 0. From Stone–Weierstrass approximation theorem 3.13, there is a polynomial P_ε: (0, 1) → such that v−P_ε∞, ≤ ε. Thus,

and we have

Since ε > 0 is arbitrary, we have (u_n, v) → 0, so that u_n 0 (weakly) on V = {v: (0, 1) → : v is continuous on [0,1] and v₂ < ∞} , with the scalar product under consideration. Notice that u_n0.

(strongly) since

We have

THEOREM 3.7.– u_n → u strongly if and only if u_n u weakly and u_n→ u when n → +∞.

PROOF.– Assume that u_n → u strongly. Then,

Thus, u_n u weakly and u_n → uwhen n → +∞. Converse is obtained by using the equality

THEOREM 3.8.– Let dim(V) = n < ∞. Then u_n → u strongly if and only if u_nu weakly.

PROOF.– Assume that u_n u weakly. Let F = {φ₁, … , φ_n} be an orthonormal basis of V. Then (u_n, φ_i) → (u, φ_i), 1 ≤ i ≤ n , so that

3.4.4. Compactness

DEFINITION 3.12.– Let S ⊂ V. S is compact if and only if any sequence {u_n: n ∈ ^*} ⊂ S has a subsequence such that . S is weakly compact if and only if any sequence {u_n: n ∈ ^*} ⊂ S has a subsequence such that .

We have:

THEOREM 3.9.– Let V be a vector space having norm . If S ⊂ V is compact then S is closed and bounded (i.e. there exists M ∈ such that u ≤ M, ∀u ∈ S).

PROOF.– Assume that S is not closed. Then, there exists a sequence {u_n: n ∈ ^*} ⊂ S, such that u_n → u (resp. u_n u) and u ∉ S. Thus, for any subsequence, . Since S is compact, u ∈ S . Then, u ∈ S and u ∉ S, what is a contradiction.

Assume that S is not bounded. Then, ∀n > 0: ∃u_n ∈ S : u_n > n . Since S is compact, {u_n: n ∈ ^*} ⊂ S has a subsequence such that . Thus, is bounded and ∃M ∈ such that. Then M > n_k, ∀k > 0. Since n_k → +∞, there is k(M) such that n_k(M) > M, so that M > n_k(M) > M, what is a contradiction.

In finite dimensional situations, we have the converse:

THEOREM 3.10.– Let V be a vector space having norm and dim(V) = n < ∞. Then S ⊂ V is compact if and only if S is closed and bounded.

PROOF.– From theorem 3.9, we have the direct implication: if S is compact, then S is closed and bounded. For the opposite, assume that S is closed and bounded. Let F = {φ₁, …, φ_n} be an orthonormal basis of V and T: V → ⁿ be the linear map T(u) = x, with x_i = (u, φ₁), 1 ≤ i ≤ n. T is bijective, since, on the one hand, ∀x ∈ ⁿ: ∃u = x₁φ₁ + …+ x_nφ_n such that T(u) = x and, on the other hand, T(u) = T(v) ⇔ (u, φ_i) = (v, φ_i), 1 ≤ i ≤ n ⇔ u = v. Moreover,

so that T(S) ⊂ ⁿis closed and bounded. As a consequence, T(S) ⊂ ⁿis compact (Heine-Borel-Lebesgue theorem. See [HOF 07], for instance).

Let has a subsequence such that. Then .

The preceding example, 3.7, shows that, contrarily to finite dimensional situations, a bounded sequence of functions does not form a compact set. In the following we will see that, for a particular class of Hibert spaces, a bounded sequence forms a weakly compact set.

3.5.1. Fixed points

A fixed point of a map, as it names explicitly mentions, is an invariant point – i.e. a point such that its image by the map is itself:

DEFINITION 3.16.– Let F: V → V be a map. We say that u ∈ V is a fixed point of F if and only if u = F(u).

A large number of numerical methods is based in the construction of fixed points, such as, for instance, Newton’s method for algebraic equations.

The following result is known as Banach’s fixed point theorem:

THEOREM 3.11.– Let V be a Hilbert space. Let F: V → V be continuous on V. F is a contraction, i.e. if there exists there exists M<1 such that

If F is a contraction, then there exists an unique u V such that u = F(u). Moreover, the sequence {u_n: n ^*} ⊂ V,u_n+1 = F(u_n) converges to u, ∀u₀ V.

PROOF.– We have

so that

and

Thus,

Let ε > 0. There exists n₀(ε) such that

Thus,

and the sequence is a Cauchy sequence. As a consequence, there exists an element u V such that u_n → u. We obtain u = F(u) from u_n+1 = F(u_n). Moreover, let v; = F(v). Then

so that (1 − M)u − v ≤ 0 ⇒ u – v = 0 ⇒ u = v.

The reader will find in the literature a large number of Fixed Point Theorems, established by Luitzen Egbertus Jan Brouwer [BRO 11], Juliusz Paweł Schauder [SHA 30], Andrei Nikolaievich Tikhonov [TIK 35] and Shizuo Kakutani [KAK 41]. Other developments may be found in [FAN 52, RIL 66, EAR 70]. A complete panorama of fixed points is given in [SMA 74].

3.6.1. Closure of a set

DEFINITION 3.17.− Let V be a vector space having scalar product (, ). Let S ⊂ V. u V is a cluster point (or adhering point) of S if and only if

Analogously, u V is a weak cluster point (or weak adhering point) of S if and only if

DEFINITION 3.18.− Let V be a vector space having scalar product (, ). Let S ⊂ V. The closure of S is denoted by and defined as:

The weak closure of S is

We have

PROPOSITION 3.3.− Let S ⊂ U. Then S ⊂ .

PROOF.− We have S ⊂ : ∀u S : u_n = u, ∀n verifies {u_n:n ^*} ⊂ S and u_n → u.

Consider {u_n: n ^*} ⊂ S such that u_n → u. Then {u_n: n ^*} ⊂ , so that u .

3.6.2. Open and closed sets

DEFINITION 3.19.− Let S ⊂ V. S is closed (respectively, weakly closed) if and only if S contains all its cluster points (resp., weak cluster points), i.e. S is open (respectively, weakly open) if and only if V − S is closed (resp. weakly closed).

We have:

THEOREM 3.12.− S is open if and only if:

PROOF.− Assume that S is open. Let us consider u S: ∀ ε > 0, ∃ v_ε S such that v_ε — u ≤ ε and v_ε ∉ S. Let u_n = v_s. We have , so that u_n → u. Moreover, u_n = v_ε ∉ S, ∀S. Thus, {u_n: n *} ⊂ V − S. Since S is open, V − S is closed. Thus, u V − S: then, we have u S and u V − S, which is a contradiction.

However, assume that

but V − S is not closed. The, ∃{u_n: n *} ⊂ V − S such that u_n → u ∉ V −S . Thus, on the one hand, u S and ∃ ε > 0 such that v − u ≤ ε ⇒ v S . On the other hand, ∃n₀(ε): n ≥ n₀(s) ⇒ u_n − u ≤ ε. Thus: u_n S, ∀n ≥ n₀(ε) and {u_n: n ε ^*} ⊂ V − S, so that ∀n ≥ n₀(ε): u_n S V − S = ø, which is a contradiction. Thus, S is closed.

Moreover:

PROPOSITION 3.4.− If S is weakly closed, then S is closed.

PROOF.− Assume that S is weakly closed. Consider {u_n: n ^*} ⊂ S such that u_n → u. From theorem 3.7, u_n u, so that u S.

PROPOSITION 3.5.− S^⊥ is weakly closed.

PROOF.− Consider {u_n:n ^*} ⊂ S^⊥ such that u_n u,. Then:

Thus, ∀v S : (u, v) = lim_n→+∞ (u_n, v) = 0, so that u S^⊥.

COROLLARY 3.2.− S^⊥⊥ = (S^⊥)^⊥ is weakly closed and .

PROOF.− S^⊥⊥ is weakly closed, from proposition 3.5: then, = S^⊥⊥. Moreover, for u S ,(u, v) = 0, ∀ vS^⊥, so that u S^⊥⊥ and we have S ⊂ S^⊥⊥. From proposition 3.3, S ⊂ ⊂ S^⊥⊥.

PROPOSITION 3.6.− Let S ⊂ V. Then is closed. If S is a vector subspace, then ⊂ V is a closed subspace.

PROOF.− See [SOU 10].

3.6.3. Dense subspaces

DEFINITION 3.20.− Let V be a vector space having scalar product (, ). Let S ⊂ V. S is dense on V if and only if .

A first example of dense subspace is furnished by the theorem of Stone-Weierstrass:

THEOREM 3.13.− Let Ω ⊂ ⁿ and F: Ω → be a continuous function. Then, ∀ε > 0, ∃ Pε, Pε polynomial, such that F − Pε ₀ ≤ ε.

PROOF.− See [DUR 12].

3.7.1. Orthogonal projection on a subspace

DEFINITION 3.21.− Let S ⊂ V. S ≠ ø. Let u ∈ V. The orthogonal projection Pu of u onto S is Pu ∈ S such that u − Pu ≤ u — v, ∀v S, i.e.

We have

THEOREM 3.14.− Let S be a closed vector subspace. Then, ∀u ∈ V, there exists an unique Pu ∈ S such that that u - Pu ≤

PROOF.– Let . Let n > 0: there exists u_n ∈ S such that

[3.1]

Thus, for m, n ≥ k,

so that

[3.2]

Since

[3.3]

we have , so that equation [3.2] implies that

[3.4]

and the sequence {u_n: n ∈ ^*} ⊂ S is Cauchy sequence. Thus, there exists Pu ∈ S (since S is closed and V is a Hilbert space) such that u_n→ Pu. Equation [3.1] shows that . Moreover, if Qu ∈ S and , then, the equality

shows that Pu – Qu² ≤ 0, so that Pu = Qu (analogous to equations [3.2], [3.3], [3.4]).

THEOREM 3.15.– Let S be a closed vector subspace and u ∈ V. Then, Pu is the orthogonal projection of u onto S if and only if

[3.5]

PROOF.– Let v ∈ S and f(λ) = u – Pu – λv². We have

Then

If Pu is the orthogonal projection of u onto S, the minimum of f is attained at λ = 0and f′ (0) = 0, so that (u – Pu, v) = 0. However, if (u – Pu, v) = 0, then f′(λ) = 0 for λ = 0, so that the minimum is attained at λ = 0 and we have u – Pu = min{u – v: v ∈ S} .

Equation [3.5] is a variational equation. It may be interpreted in terms of orthogonality: vector u – Pu is orthogonal to all the elements of the subspace S. A simple geometrical interpretation is given in Figure 3.5: in dimension 2, a vector subspace is a line passing through the origin.

One of the main consequences of the orthogonal projection is [RSZ 07]

THEOREM 3.16.– Let S ⊂ V be a vector subspace. S is dense on V if and only if S^⊥ = {0}.

PROOF.– Let S be dense on V and v ∈ S^⊥ Then, on the one hand, there exists a sequence {v_n: n ∈ ^*} ⊂ V such that v_n → v; on the other hand, (v_n, v) = 0, ∀n ∈ *. Thus, from theorem 3.7, (v, v) = 0 ⇒ v = 0. Thus, S^⊥ = {0}.

Let S^⊥ = {0}. Assume that . Then, there exists u ∈ V, u ∉ . Since is closed, is open: from theorem 3.12, there is ε > 0 such that . Let Pu be the orthogonal projection of u onto : equation [3.5] shows that u – . Since S ⊂ , we have u – Pu ∈ S^⊥ Thus, u — Pu = 0, so that u = Pu ∈ . Thus, u ∉ and u ∉ , what is a contradiction.

3.7.2. Orthogonal projection on a convex subset

DEFINITION 3.22.– Let S ⊂ V. S ≠ Ø. S is convex if and only if

THEOREM 3.17.– Let S be a closed non-empty convex. Then, ∀u ∈ V, there exists a unique Pu ∈ S such that that u – Pu ≤ u – v, ∀v , ∈ S. Moreover, Pu is the orthogonal projection of u onto S if

[3.6]

PROOF.– The proof of existence and uniqueness is analogous to these given for theorem 3.14. Since S is convex, equations [3.2], [3.3], [3.4] are verified and the conclusion is obtained by the same way (the sequence is a Cauchy sequence, V is Hilbert, S is closed). The inequality is obtained as follows: let v ∉ S and f(λ) = u – Pu – λ{v-Pu)². Since

we have f(0) ≤ f(λ), ∀λ ∈ (0, 1), so that f′(0) = -2(u – Pu, v – Pu) ≥ 0 and we have the inequality.

Equation [3.6] is a variational inequality. It may be interpreted in terms of orthogonality: vector u – Pu is orthogonal to the tangent space of set S. A simple geometrical interpretation in dimension 2 is given in Figure 3.6: we see that the angle between u – Pu and v – Pu is superior to π/2.

3.7.3. Orthogonal projection on an affine subspace

DEFINITION 3.23.– Let S ⊂ V. S ≠ Ø. S is an affine subspace if and only if

THEOREM 3.18.– Let S be a closed non-empty affine subspace. Then, ∀u V, there exists an unique Pu ∈ S such that that u – Pu ≤ u – v, ∀v ∈ S. Moreover, Pu is the orthogonal projection of u onto S if and only if

[3.7]

PROOF.– Notice that S is a closed convex set, so that theorem 3.17 applies. In order to prove the equality, notice that, for v ∈ S, 2Pu – v ∈ S, so that we have

Thus, (u – Pu, v – Pu) ≤ 0 and (u – Pu, v – Pu) ≥ 0, which establishes the equality.

Equation [3.7] is a variational equation. It may be interpreted in terms of orthogonality: vector u – Pu is orthogonal to the differences of elements of S. A simple geometrical interpretation in dimension 2 is given in Figure 3.7: we see that u – Pu is orthogonal to v – Pu.

Affine subspaces are generally defined by introducing a translation of a vector subspace:

In this case

THEOREM 3.19.– Let S = S₀ + {u₀}, where S₀ is a closed vector subspace and u₀ ∈ V. Then, ∀u ∈ V, there exists an unique Pu ∈ S such that that u — Pu ≤ u — v, ∀v ∈ S. Moreover, Pu is the orthogonal projection of u onto S if and only if

[3.8]

PROOF.– Notice that S is a closed affine subspace, so that theorem 3.18 applies. For the equality, notice that, for any W ∈ S₀: W + Pu ∈ S. Indeed, U = Pu – u₀ ∈ S₀ and, as a consequence, w + U ∈ S₀, so that w + Pu = w + U + u₀ ∈ S. Thus, equation [3.7] is equivalent to equation [3.8].

Equation [3.8] is a variational equation. A simple geometrical interpretation in dimension 2 is given in Figure 3.8: we see that u – Pu is orthogonal to S₀ parallel to S, which is generated by a translation of S₀.

3.7.4. Matlab® determination of orthogonal projections

Orthogonal projections may be assimilated to linear equations and determined into a way analogous to the one introduced in section 2.2.4.

Let us consider the situation where S is a subspace: we may consider a basis and look for an approximation

Then, the variational equation [3.5] yields that

Thus, the coefficients U = (u₁, …, u_n)^t satisfy a linear system AU = B, with and .

When using the scalar product sp0, defined in section 3.2.5, and the basis defined in section 2.2.6, the coefficients U may be determined by using the method approx_coeffs. Notice that matrix A corresponds to methods varmat_mean, varmat_int, while vector B corresponds to methods sm_mean and sm_int. The method of evaluation is selected by _ choosing ‘variational_mean’, ‘variational_int’ or ‘variational_sp’.

Otherwise, matrix A and vector B may be determined by using the class introduced in section 3.2.5. We introduce below a new class corresponding to this last situation. Analogously to the approach presented in 3.2.6, we assume that basis{i} defines as a structure having properties dim, dimx, component, where component is a cell array such that each element is a structure having as properties value and grad. u is assumed to be analogously defined. In this case, an example of class for orthogional projection is given in program 3.6. The class contains two methods: coeffs, which evaluates the coefficients of the orthogonal projection and funct, which generates a subprogram (anonymous function) evaluating the projection.

EXAMPLE 3.8.– Let us consider the family 4 and the function u(x) = x⁵ on (−1, 1). Using the scalar product sp0, the orthogonal projection is . The code

xlim.lower.x = -1;
xlim.upper.x = 1;
xlim.dim = 1;
u.dim = 1;
u.dimx = 1;
u.component = cell(u.dim,1);
u.component{1}.value = @(x) x^5;
u.component{1}.grad = @(x) 5*x^4;
sp_space = @(u,v)
scalar_product.sp0(u,v,xlim,'subprogram');
c = orthogonal_projection.coeffs(u,basis,sp_space);
pu = orthogonal_projection.funcproj(c,basis);

generates a subprogram pu which evaluates the orthogonal projection. Analogously,

U.dim = 1;
U.dimx = 1;
U.component = cell(U.dim,1);
U.component{1}.value =
spam.partition(u.component{1}.value,p);
U.component{1}.grad = cell(U.dim,1);
U.component{1}.grad{1} =
spam.partition(u.component{1}.grad,p);
sp_space = @(u,v)
scalar_product.sp0(u,v,p,'table');
c_tab =
orthogonal_projection.coeffs(U,tb,sp_space);
pu_tab =
orthogonal_projection.funcproj(c_tab,basis);

generates a subprogram pu_tab which evaluates the orthogonal projection. The results are shown in Figure 3.9.

EXAMPLE 3.9.– Let us consider the family and the function u(x) = sin(x) on (0, 2π:). Using the scalar product sp0, the orthogonal projection is . The code

xlim.lower.x = 0;
xlim.upper.x = 2*pi;
xlim.dim = 1;
u.dim = 1;
u.dimx = 1;
u.component = cell(u.dim,1);
u.component{1}.value = @(x) sin(x);
u.component{1}.grad = @(x) cos(x);

defines u. pu and pu_tab are generated analogously to example 3.3. The results are_ shown in Figure 3.10.

3.8.1. Series

DEFINITION 3.24.– A series of elements of V is a sequence {S_n: n ∈ ^*}, such that . If S_n→ v ∈ V, we say that the series is convergent and its sum is . A series that is not convergent is said to be divergent.

S_n is usually referred to as a partial sum. The series

is the series of the residual of order n. R_n is defined by the sequence

We have

THEOREM 3.20.– The following assertions are equivalent

PROOF.– Assume (i). Let . Then . Theorem 3.5 shows that all the subsequences of converge to the same limit. Thus, for n > 1: R_{n, n+k} = S_n+k – , so that we have (ii).

Assume (ii). Then (iii) is immediate.

Assume (iii). We have . Taking the limit for k → +∞, we have S_n−1 + R_n = S_m−1 + R_m. Thus, s = S_n−1 + R_n is independent of n and we have ∀n, k > 1: S_n+k = S_n–1 + R_{n, k} → s, for k → +∞. Thus

what shows that S_k+1 → s, so that converges.

3.8.2. Separable spaces and Fourier series

DEFINITION 3.25.– Let and [F] be the set of the finite linear combinations of elements of F:

F is a total family if and only if , i.e. [F] is dense on V. We say that F is a Hilbert basis if and only if F is orthonormed and total.

DEFINITION 3.26.– V is separable if and only if there exists F ⊂ V such that F is total.

We have:

THEOREM 3.21.– Let be a Hilbert basis of V. Let u, v ∈ V. Then

1. i) is a closed subspace.
2. ii) The orthogonal projection
3. iii) strongly in V.
4. iv)
5. v)
6. vi) converges.

PROOF.– See [SOU 10].

is the Fourier series of u associated with the Hilbert basis F. Equality 3.21. (iv) is known as Bessel-Parseval equality.

Separable spaces have the weak Bolzano–Weierstrass property:

THEOREM 3.22.– Let V be a separable Hilbert space. Let {u_n: n ∈ ^*} ⊂ V be a bounded sequence (∃M ∈ : ∀n ∈ * : u_n ≤ M). Then {u_n:n ∈ ^*} has a subsequence which is weakly convergent.

PROOF.– See [SOU 10].

3.9.1. Linear functionals

DEFINITION 3.27.– A linear functional is an application ℓ: V → such that

A linear functional ℓ verifies ℓ(0) = 0 : since 0 = 2 × 0, we have ℓ(0) = ℓ(2 × 0). Thus, ℓ(0) = 2 × ℓ(0) and ℓ(0) = 0.

DEFINITION 3.28.– The norm of a linear functional ℓ is

We have

PROPOSITION 3.7.–

and

THEOREM 3.23.– Let ℓ:V → be a linear functional. ℓ is continuous if and only if ℓ < ∞.

PROOF .-Assume that ℓ < ∞ Then

so that ℓ satisfies Lipschitz’s condition and, thus, is continuous (observe that, for u_n → u, we have .

Assume that ℓ is continuous. Let us establish that ℓ < ∞.

Then

But

Thus, 0 > +∞, what is a contradiction.

COROLLARY 3.3.– Let ℓ: V → be a linear functional. ℓ is continuous if and only if there exists M ∈ such that In this case, ℓ ≤ M .

PROOF.– Assume that ℓ is continuous. Then M = ℓ verifies

Assume that there exists M ∈ such that

and ℓ is continuous.

EXAMPLE 3.10.– Let us give an example of a linear functional ℓ which is not continuous and has ℓ = ∞. Let us consider

Let ℓ:V → be given by (v) = v(0). Let us consider the sequence

Then, for 0 < α < 1/2,

while

Assume that there exists M ∈ such that

So that

what is a contradiction, since M ∈ . Thus, M ∈ such that ℓ(v) ≤ M v, ∀v ∈ V: ℓ is not continuous and ℓ = ∞. This functional is kwown as Dirac’s delta or Dirac’s mass (see Chapter 6).

An important property of continuous linear functional is the continuous extension:

THEOREM 3.24.– Let S ⊂ V be a dense subspace. Let ℓ : S → be such that there exists M ∈ such that ∀v ∈ S: ℓ(v) ≤ M v. In this case, ℓ has an extension ℓ V → such that ℓ ≤ M .

PROOF.– Let v ∈ V Since S is dense on V, there is a sequence {v_n: n ∈ ^*} ⊂ S such that v_n → v. Let us consider the sequence of real numbers {ℓ(v_n): n ∈ ^*} ⊂ . This is a Cauchy sequence, since

Then, ℓ(v_n) → λ ∈ . Let {w_n: n ∈ ^*} ⊂ S such that w_n → v and {ℓ(v_n) → η ∈ . Then.

So that λ = η. Thus, we may define

ℓ is linear: if S ∋ v_n → v ∈ V, S ∋ u_n → u ∈ V, α ∈ , then

so that ℓ(i + av) = ℓ(u) + aℓ(v). Finally, by taking the limit for n → ∞,

DEFINITION 3.29.– The topological dual space V’ associated with V is

3.9.2. Kernel of a linear functional

DEFINITION 3.30.– The Kernel of a linear functional ℓ is

Notice that Ker (ℓ) is a subspace, since

and, since ℓ(αu + i) = αℓ(u) + ℓ(v),

We have

PROPOSITION 3.8.– Let ℓ: V → be a linear functional. ℓ is continuous if and only if Ker(ℓ) is closed.

PROOF.– Assume that ℓ is continuous. Then

so that

and

Thus Ker(ℓ) is closed.

Assume that Ker(ℓ) is closed. Let us establish that ℓ < ∞. This is immediate if ℓ = 0, since, in this case, ℓ = 0. Let us assume ℓ ≠ 0 and ≠ = ∞. Then, on the one hand,

and, on the other hand, there exists ū such that ℓ(ū) = 1. Let

Then

so that {u_n}_n>0 ⊂ Ker(ℓ). Moreover,

so that u_n → ū. Since Ker(ℓ) is closed, we have ū ∈ iver(0) ⇒ ℓ(ū) = 0. Thus, 1 = ℓ(ū) = 0, what is a contradiction.

3.9.3. Riesz’s theorem

The main result is the Riesz’s theorem [RSZ 09, FRE 07a]:

THEOREM 3.25.– Let ℓ: V → be a continuous linear functional. Then there exists one and only one u ∈ V such that:

Moreover

PROOF.– The result is immediate when ℓ = 0, since, in this case, ℓ = 0 and u = 0. Let us assume ℓ ≠ 0. Then, there exists w such that ℓ(w) = 1. Let us denote Pw its orthogonal projection onto Ker(ℓ). We have w – Pw ≠ 0, since ℓ(w) = 1 ≠ 0 = ℓ(Pw). Let us consider,

From the properties of the orthogonal projection, we have u ⊥ Ker(ℓ), so that, (u, Pw) = 0. Moreover,

Let v ∈ V: we have,

Since ℓ(v − ℓ(v)w) = ℓ(v) − ℓ(v)ℓ(w) = ℓ(v) ℓ(v) = 0, we have v − ℓ(v)w ∈ Ker(ℓ) and

Moreover,

and

Riesz’s theorem establishes a connection between linear functionals and scalar products. Since scalar products represent the virtual work of internal efforts and linear functional represent the virtual work of external forces, Riesz’s theorem may be interpreted as a connection between the work of external forces and the variation of internal energy, which corresponds to the work of internal efforts. More precisely, Riesz’s theorem connects the virtual work of external forces to the virtual work of internal efforts.

Moreover, Riesz’s theorem connects a linear functional and a field: it may also be interpreted as the connection between a virtual work and a force. This connection is formalized by Riesz canonical isometry:

Riesz’s theorem shows that ∏ is a bijection, so that ∏⁻¹ exists. Moreover, on the one hand, ∏(u) = u, while, on the other hand, ∏⁻¹(ℓ) = ℓ.

3.10.1. 1D situations

For instance, let us consider Ω = (a, b) and the space V = L²(Ω). The operator defined by:

is linear, bounded, compact and self-adjoint. Let ϕ be an eigenvector of T, associated with the eigenvalue α:

We have:

Let us observe that α_n ≠ 0, since,

which is a contradiction. Thus, denoting λ = 1/α, we have:

In this case, the solution is a trigonometric family:

3.10.2. 2D situations

Analogously, if Ω ⊂ ² is a regular bounded domain, H = L²(Ω),

is also linear, bounded, compact and self-adjoint Let ϕ be an eigenvector of T, associated with the eigenvalue α:

We have:

Here again, α_n ≠ 0, since,

what is a contradiction? Thus, denoting λ = 1/α, we have:

In this case, the solution is not explicitly known for general domains: it is explicitly known only for particular shapes of Ω. Otherwise, it must be numerically determined. For special geometries, the eigenfunctions are often determined by separation of variables.

3.10.2.1. The case of a rectangle

If , we may consider (x, y) = X(x)Y(y).

Then,

Thus, we have,

So,

and:

Moreover,

Thus, we have:

So,

The solution is a product of trigonometric families: X = u_m and Y = u_n are solutions of the 1D case,

3.10.2.2. The case of a disk

If Ω= {(x, y): x²+y² < α²}, we may consider (x, y) = R(r) Θ (θ), = rcosθ, y = rsin θ. Let us denote by μ′ the derivative of a function μ(ξ) with respect to its single real variable ξ ∈ − for instance R′ = dR/dr, Θ′ = dΘ/dθ, etc. Since

we have:

and

So,

Thus,

Let us introduce : we have (recall that y′ = dy/dp, while R′ = dR/dr)

and

For β = −m², this is the classical Bessel’s equation and Θ is a trigonometric function. Thus, denoting by J_m the m-th Bessel’s function and by z_mn its n-th zero:

3.10.3. 3D situations

If Ω ⊂ ³ is a regular bounded domain, H = L² (Ω),

is again linear, bounded, compact and self-adjoint. Thus, an eigenvector ø, associated with the eigenvalue α verifies, with λ = 1/α,

Here, the solution is explicitly known only for particular shapes of Ω and it is often determined by separation of variables.

3.10.3.1. The case of a rectangular parallelepiped

If , we may consider Then, we obtain:

3.10.3.2. The case of a cylinder

, we may consider . Since,

we have:

Thus,

So,

For, , we take and we obtain, analogously to the 2D-case,

Thus, y is again the solution of Bessel’s equation, while Z and Θ are trigonometric functions and we have:

3.10.3.3. The case of a ball

For Ω = {(x, y, z) : x² + y² + z² < α²}, we may consider rcosφ associated with the eigenvalue α. Since,

we have, for λ = 1/α,

and

For , we have (recall that y′ = dy/ds, while )

and

Thus

and

This equation is known as Legendre’s equation. The solutions are the associated Legendre functions . Moreover, Θ is a trigonometric function and verifies

Taking again , we have:

and

The solutions of this equation are the spherical Bessel’s functions. Thus, the eigenfunctions are:

z_m+1/2,n is the n-th zero of J_m+1/2. The angular term is called a spherical harmonic.

3.10.4. Using a sequence of finite families

An alternative to the generation of a hilbertian basis consists of the use of a family of finite families, i.e. F ⊂ V such that,

For instance, families of finite elements may be used: in such a case, F_n corresponds to a particular discretization (i.e. a particular mesh) and F is the family of all the possible discretizations (i.e. of all the possible meshes). Finite volume approximations or wavelets may be interpreted by the same way.