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In mathematics, more precisely in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, an energetic space is, intuitively, a subspace of a given
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
equipped with a new "energetic"
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. The motivation for the name comes from
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, as in many physical problems the
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
of a system can be expressed in terms of the energetic inner product. An example of this will be given later in the article.


Energetic space

Formally, consider a real Hilbert space X with the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
(\cdot, \cdot) and the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
\, \cdot\, . Let Y be a linear subspace of X and B:Y\to X be a strongly monotone
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
, that is, a linear operator satisfying * (Bu, v)=(u, Bv)\, for all u, v in Y * (Bu, u) \ge c\, u\, ^2 for some constant c>0 and all u in Y. The energetic inner product is defined as :(u, v)_E =(Bu, v)\, for all u,v in Y and the energetic norm is :\, u\, _E=(u, u)^\frac_E \, for all u in Y. The set Y together with the energetic inner product is a
pre-Hilbert space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. The energetic space X_E is defined as the completion of Y in the energetic norm. X_E can be considered a subset of the original Hilbert space X, since any
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
in the energetic norm is also Cauchy in the norm of X (this follows from the strong monotonicity property of B). The energetic inner product is extended from Y to X_E by : (u, v)_E = \lim_ (u_n, v_n)_E where (u_n) and (v_n) are sequences in ''Y'' that converge to points in X_E in the energetic norm.


Energetic extension

The operator B admits an energetic extension B_E :B_E:X_E\to X^*_E defined on X_E with values in the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
X^*_E that is given by the formula :\langle B_E u , v \rangle_E = (u, v)_E for all u,v in X_E. Here, \langle \cdot , \cdot \rangle_E denotes the duality bracket between X^*_E and X_E, so \langle B_E u , v \rangle_E actually denotes (B_E u)(v). If u and v are elements in the original subspace Y, then :\langle B_E u , v \rangle_E = (u, v)_E = (Bu, v) = \langle u, B, v\rangle by the definition of the energetic inner product. If one views Bu, which is an element in X, as an element in the dual X^* via the
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the ...
, then Bu will also be in the dual X_E^* (by the strong monotonicity property of B). Via these identifications, it follows from the above formula that B_E u= Bu. In different words, the original operator B:Y\to X can be viewed as an operator B:Y\to X_E^*, and then B_E:X_E\to X^*_E is simply the function extension of B from Y to X_E.


An example from physics

Consider a
string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
whose endpoints are fixed at two points a on the real line (here viewed as a horizontal line). Let the vertical outer
force density In fluid mechanics, the force density is the negative gradient of pressure. It has the physical dimensions of force per unit volume. Force density is a vector field representing the flux density of the hydrostatic force within the bulk of a fl ...
at each point x (a\le x \le b) on the string be f(x)\mathbf, where \mathbf is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction ve ...
pointing vertically and f:
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline of ...
to \mathbb R. Let u(x) be the
deflection Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflectio ...
of the string at the point x under the influence of the force. Assuming that the deflection is small, the
elastic energy Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, ...
of the string is : \frac \int_a^b\! u'(x)^2\, dx and the total
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of the string is : F(u) = \frac \int_a^b\! u'(x)^2\,dx - \int_a^b\! u(x)f(x)\,dx. The deflection u(x) minimizing the potential energy will satisfy the
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
: -u''=f\, with
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
:u(a)=u(b)=0.\, To study this equation, consider the space X=L^2(a, b), that is, the
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
of all
square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute val ...
s u:
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline of ...
to \mathbb R in respect to the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
. This space is Hilbert in respect to the inner product : (u, v)=\int_a^b\! u(x)v(x)\,dx, with the norm being given by : \, u\, =\sqrt. Let Y be the set of all twice continuously differentiable functions u:
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline of ...
to \mathbb R with the
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
u(a)=u(b)=0. Then Y is a linear subspace of X. Consider the operator B:Y\to X given by the formula : Bu = -u'',\, so the deflection satisfies the equation Bu=f. Using
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
and the boundary conditions, one can see that : (Bu, v)=-\int_a^b\! u''(x)v(x)\, dx=\int_a^b u'(x)v'(x) = (u, Bv) for any u and v in Y. Therefore, B is a symmetric linear operator. B is also strongly monotone, since, by the
Friedrichs's inequality In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the ''Lp'' norm of a function using ''Lp'' bounds on the weak derivatives of the function and the geometry of the domain, and ...
: \, u\, ^2 = \int_a^b u^2(x)\, dx \le C \int_a^b u'(x)^2\, dx = C\,(Bu, u) for some C>0. The energetic space in respect to the operator B is then the
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
H^1_0(a, b). We see that the elastic energy of the string which motivated this study is : \frac \int_a^b\! u'(x)^2\, dx = \frac (u, u)_E, so it is half of the energetic inner product of u with itself. To calculate the deflection u minimizing the total potential energy F(u) of the string, one writes this problem in the form :(u, v)_E=(f, v)\, for all v in X_E. Next, one usually approximates u by some u_h, a function in a finite-dimensional subspace of the true solution space. For example, one might let u_h be a continuous piecewise linear function in the energetic space, which gives the
finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
. The approximation u_h can be computed by solving a system of linear equations. The energetic norm turns out to be the natural norm in which to measure the error between u and u_h, see Céa's lemma.


See also

*
Inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
*
Positive-definite kernel In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving ...


References

* *{{cite book , last = Johnson , first = Claes , title = Numerical solution of partial differential equations by the finite element method , publisher = Cambridge University Press , date = 1987 , pages = , isbn = 0-521-34514-6 Functional analysis Hilbert space