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:
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* ''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
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 ...
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 ...
. Let
be a linear subspace of
and
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
*
for all
in
*
for some constant
and all
in
The energetic inner product is defined as
:
for all
in
and the energetic norm is
:
for all
in
The set
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
is defined as the
completion of
in the energetic norm.
can be considered a subset of the original Hilbert space
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
(this follows from the strong monotonicity property of
).
The energetic inner product is extended from
to
by
:
where
and
are sequences in ''Y'' that converge to points in
in the energetic norm.
Energetic extension
The operator
admits an energetic extension
:
defined on
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 ...
that is given by the formula
:
for all
in
Here,
denotes the duality bracket between
and
so
actually denotes
If
and
are elements in the original subspace
then
:
by the definition of the energetic inner product. If one views
which is an element in
as an element in the dual
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
will also be in the dual
(by the strong monotonicity property of
). Via these identifications, it follows from the above formula that
In different words, the original operator
can be viewed as an operator
and then
is simply the function extension of
from
to
An example from physics

Consider a
string
String or strings may refer to:
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Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
whose endpoints are fixed at two points