|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| :''This article deals with [[linear map]]s from a [[vector space]] to its field of [[scalar (mathematics)|scalar]]s. These maps '''may''' be [[functional (mathematics)|functionals]] in the traditional sense of functions of functions, but this is not necessarily the case.''
| | Perfect lifetime is 1 which is free from any physical anguish. A individual experiencing pain caused by hemorrhoids could yearn for several treatment for hemorrhoids to ease his pain. However, nothing is more uplifting than to realize that you needn't undergo any kind of medicine or operation. In fact, there are a amount of signifies to avoid the appearance of hemorrhoids plus these preventive procedures are awfully simple.<br><br>The first [http://hemorrhoidtreatmentfix.com/hemorrhoid-symptoms symptoms of hemorrhoids] is to utilize creams plus ointments. These lotions plus ointments is used found on the outer rectal region inside order to aid relaxing blood vessels. This can lower the inflammation because creams plus ointments will relax the tissue. But, this kind of treatment is considered to be wise for helping in really a short period. It is fairly possible that the hemorrhoid will probably to happen again.<br><br>There are many treatments that could be utilized for hemorrhoid. The first and the most popular is the cream and ointment. These are to be rubbed onto the affected piece of the anus. It helps to soothe the already inflamed blood vessels plus a momentary relief is accomplished. There is a relaxation of the tissues of the rectal column thus far the hemorrhoid is not thus much bulged. If there is a bulge yet, the pain relief may not do so much to aid.<br><br>Having a advantageous bowel movement etiquette is without doubt one of the right methods to avoid hemorrhoids. Everytime you feel the urge, do thus without hindrance. Numerous delays of going to the rest space brings about constipation that in turn causes you to strain difficult inside order to eliminate a bowels. As rapidly because you're finished, immediately receive up. Staying too long found on the bathroom seat may amplify pressure on your veins. After the bowel movement, you may moreover try sitting inside a tub with lukewarm water for 10 to 15 minutes to relax the rectum.<br><br>Right now, there are a great deal of hemorrhoid treatments. And yes, there are the painless hemorrhoid treatments furthermore available. Examples of such as use of petroleum jelly, the utilization of ointment phenylephrine or Preparation H, and even the easy utilize of soft cotton underwear. These are typically painless for with them we don't should go under the knife.<br><br>One of the number one methods of reducing yourself within the pain and itchiness is to soak in a warm Sitz bath. Do this for regarding 10 to 15 minutes, many instances a day. Warm water relaxes the muscles and reduces the swelling. You are able to additionally apply coconut oil to the affected region. It usually have a soothing impact though it won't last which lengthy. Applying witch hazel could equally aid soothe the pain plus itch. It decreases the bleeding and minimizes the swelling.<br><br>Whenever this shower is performed, we may then like to take some ice (wrapped up in a cloth) plus apply it to a anal region. This causes a decrease in the amount of blood which is flowing to the area; also ice may have a numbing effect giving we a small more relief from pain. |
| | |
| In [[linear algebra]], a '''linear functional''' or '''linear form''' (also called a '''[[one-form]]''' or '''covector''') is a [[linear map]] from a [[vector space]] to its field of [[scalar (mathematics)|scalar]]s. In [[Euclidean space|'''R'''<sup>''n''</sup>]], if [[euclidean vector|vectors]] are represented as [[column vector]]s, then linear functionals are represented as [[row vector]]s, and their action on vectors is given by the [[dot product]], or the [[matrix product]] with the [[row vector]] on the left and the [[column vector]] on the right. In general, if ''V'' is a [[vector space]] over a [[field (mathematics)|field]] ''k'', then a linear functional ''f'' is a function from ''V'' to ''k'', which is linear: | |
| | |
| :<math>f(\vec{v}+\vec{w}) = f(\vec{v})+f(\vec{w})</math> for all <math>\vec{v}, \vec{w}\in V</math>
| |
| :<math>f(a\vec{v}) = af(\vec{v})</math> for all <math>\vec{v}\in V, a\in k.</math>
| |
| | |
| The set of all linear functionals from ''V'' to ''k'', Hom<sub>''k''</sub>(''V'',''k''), is itself a vector space over ''k''. This space is called the [[dual space]] of ''V'', or sometimes the '''algebraic dual space''', to distinguish it from the [[continuous dual space]]. It is often written ''V*'' or ''V′'' when the field ''k'' is understood. | |
| | |
| == Continuous linear functionals ==
| |
| {{see also|Continuous linear operator}}
| |
| | |
| If ''V'' is a [[topological vector space]], the space of [[continuous function|continuous]] linear functionals — the ''[[continuous dual space|continuous dual]]'' — is often simply called the dual space. If ''V'' is a [[Banach space]], then so is its (continuous) dual. To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the ''algebraic dual''. In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, although this is not true in infinite dimensions.
| |
| | |
| == Examples and applications ==
| |
| === Linear functionals in R<sup>''n''</sup> ===
| |
| Suppose that vectors in the real coordinate space '''R'''<sup>''n''</sup> are represented as column vectors
| |
| | |
| :<math>\vec{x} = \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math>
| |
| | |
| Then any linear functional can be written in these coordinates as a sum of the form:
| |
| | |
| :<math>f(\vec{x}) = a_1x_1 + \cdots + a_n x_n.</math>
| |
| | |
| This is just the matrix product of the row vector [''a''<sub>1</sub> ... ''a''<sub>''n''</sub>] and the column vector <math>\vec{x}</math>:
| |
| :<math>f(\vec{x}) = [a_1 \dots a_n] \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math>
| |
| | |
| === Integration ===
| |
| Linear functionals first appeared in [[functional analysis]], the study of [[function space|vector spaces of functions]]. A typical example of a linear functional is [[integral|integration]]: the linear transformation defined by the [[Riemann integral]]
| |
| | |
| :<math>I(f) = \int_a^b f(x)\, dx</math>
| |
| | |
| is a linear functional from the vector space C[''a'',''b''] of continuous functions on the interval [''a'', ''b''] to the real numbers. The linearity of ''I''(''f'') follows from the standard facts about the integral:
| |
| :<math>I(f+g) = \int_a^b(f(x)+g(x))\, dx = \int_a^b f(x)\, dx + \int_a^b g(x)\, dx = I(f)+I(g)</math>
| |
| :<math>I(\alpha f) = \int_a^b \alpha f(x)\, dx = \alpha\int_a^b f(x)\, dx = \alpha I(f).</math>
| |
| | |
| === Evaluation ===
| |
| Let ''P<sub>n</sub>'' denote the vector space of real-valued polynomial functions of degree ≤''n'' defined on an interval [''a'',''b'']. If ''c'' ∈ [''a'', ''b''], then let ev<sub>''c''</sub> : ''P<sub>n</sub>'' → '''R''' be the '''evaluation functional''':
| |
| :<math>\operatorname{ev}_c f = f(c).</math>
| |
| The mapping ''f'' → ''f''(''c'') is linear since
| |
| :<math>(f+g)(c) = f(c) + g(c)</math>
| |
| :<math>(\alpha f)(c) = \alpha f(c).</math>
| |
| | |
| If ''x''<sub>0</sub>, ..., ''x<sub>n</sub>'' are ''n''+1 distinct points in [''a'',''b''], then the evaluation functionals ev''<sub>x<sub>i</sub></sub>'', ''i''=0,1,...,''n'' form a [[basis of a vector space|basis]] of the dual space of ''P<sub>n</sub>''. ({{harvtxt|Lax|1996}} proves this last fact using [[Lagrange interpolation]].)
| |
| | |
| === Application to quadrature ===
| |
| The integration functional ''I'' defined above defines a linear functional on the [[linear subspace|subspace]] ''P<sub>n</sub>'' of polynomials of degree ≤ ''n''. If ''x''<sub>0</sub>, …, ''x''<sub>''n''</sub> are ''n''+1 distinct points in [''a'',''b''], then there are coefficients ''a''<sub>0</sub>, …, ''a''<sub>''n''</sub> for which
| |
| | |
| :<math>I(f) = a_0 f(x_0) + a_1 f(x_1) + \dots + a_n f(x_n)</math>
| |
| | |
| for all ''f'' ∈ ''P''<sub>''n''</sub>. This forms the foundation of the theory of [[numerical quadrature]].
| |
| | |
| This follows from the fact that the linear functionals ''ev<sub>x<sub>i</sub></sub>'' : ''f'' → ''f''(''x''<sub>''i''</sub>) defined above form a [[basis of a vector space|basis]] of the dual space of ''P''<sub>''n''</sub> {{harv|Lax|1996}}.
| |
| | |
| === Linear functionals in quantum mechanics ===
| |
| Linear functionals are particularly important in [[quantum mechanics]]. Quantum mechanical systems are represented by [[Hilbert space]]s, which are [[antilinear|anti]]-[[linear isomorphism|isomorphic]] to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see [[bra-ket notation]].
| |
| | |
| === Distributions ===
| |
| In the theory of [[generalized function]]s, certain kinds of generalized functions called [[distribution (mathematics)|distributions]] can be realized as linear functionals on spaces of [[test function]]s.
| |
| | |
| == Properties ==
| |
| * Any linear functional ''L'' is either trivial (equal to 0 everywhere) or [[surjective]] onto the scalar field. Indeed, this follows since just as the image of a vector [[linear subspace|subspace]] under a linear transformation is a subspace, so is the image of ''V'' under ''L''. But the only subspaces (i.e., ''k''-subspaces) of ''k'' are {0} and ''k'' itself.
| |
| | |
| * A linear functional is continuous if and only if its [[Kernel (linear operator)|kernel]] is closed {{harv|Rudin|1991|loc=Theorem 1.18}}.
| |
| | |
| * Linear functionals with the same kernel are proportional.
| |
| | |
| * The absolute value of any linear functional is a [[seminorm]] on its vector space.
| |
| | |
| ==Visualizing linear functionals==
| |
| [[File:Gradient 1-form.svg|thumb|160px|Geometric interpretation of a 1-form '''α''' as a stack of [[hyperplane]]s of constant value, each corresponding to those vectors that '''α''' maps to a given scalar value shown next to it along with the "sense" of increase. The zero plane ('''<span style="color:purple;">purple</span>''') is through the origin.]]
| |
| | |
| In finite dimensions, a linear functional can be visualized in terms of its [[level set]]s. In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel [[hyperplane]]s. This method of visualizing linear functionals is sometimes introduced in [[general relativity]] texts, such as [[Gravitation (book)|Gravitation]] by {{harvtxt|Misner|Thorne|Wheeler|1973}}.
| |
| {{-}}
| |
| | |
| == Dual vectors and bilinear forms ==
| |
| {{See also|Hodge dual}}
| |
| [[File:1-form linear functional.svg|thumb|400px|Linear functionals (1-forms) '''α''', '''β''' and their sum '''σ''' and vectors '''u''', '''v''', '''w''', in [[three-dimensional space|3d]] [[Euclidean space]]. The number of (1-form) [[hyperplane]]s intersected by a vector equals the [[inner product]].<ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|page=57|isbn=0-7167-0344-0}}</ref>]]
| |
| | |
| Every non-degenerate [[bilinear form]] on a finite-dimensional vector space ''V'' gives rise to an [[isomorphism]] from ''V'' to ''V*''. Specifically, denoting the bilinear form on ''V'' by < , > (for instance in [[Euclidean space]] <''v'',''w''> = ''v''•''w'' is the [[dot product]] of ''v'' and ''w''), then there is a natural isomorphism <math>V\to V^*:v\mapsto v^*</math> given by
| |
| | |
| : <math> v^*(w) := \langle v, w\rangle.</math>
| |
| | |
| The inverse isomorphism is given by <math>V^* \to V : f \mapsto f^* </math> where ''f*'' is the unique element of ''V'' for which for all ''w'' ∈ ''V''
| |
| | |
| : <math> \langle f^*, w\rangle = f(w).</math>
| |
| | |
| The above defined vector ''v''* ∈ ''V*'' is said to be the '''dual vector''' of ''v'' ∈ ''V''.
| |
| | |
| In an infinite dimensional [[Hilbert space]], analogous results hold by the [[Riesz representation theorem]]. There is a mapping ''V'' → ''V*'' into the ''continuous dual space'' ''V*''. However, this mapping is [[antilinear]] rather than linear.
| |
| {{-}}
| |
| | |
| ==Bases in finite dimensions==
| |
| | |
| ===Basis of the dual space in finite dimensions===
| |
| Let the vector space ''V'' have a basis <math>\vec{e}_1, \vec{e}_2,\dots,\vec{e}_n</math>, not necessarily [[orthogonal]]. Then the [[dual space]] ''V*'' has a basis <math>\tilde{\omega}^1,\tilde{\omega}^2,\dots,\tilde{\omega}^n</math> called the [[dual basis]] defined by the special property that
| |
| | |
| :<math> \tilde{\omega}^i (\vec e_j) = \left\{\begin{matrix} 1 &\mathrm{if}\ i=j\\ 0 &\mathrm{if}\ i\not=j.\end{matrix}\right. </math>
| |
| | |
| Or, more succinctly,
| |
| | |
| :<math> \tilde{\omega}^i (\vec e_j) = \delta^i_j </math>
| |
| | |
| where δ is the [[Kronecker delta]]. Here the superscripts of the basis functionals are not exponents but are instead [[covariance and contravariance|contravariant]] indices.
| |
| | |
| A linear functional <math>\tilde{u}</math> belonging to the dual space <math>\tilde{V}</math> can be expressed as a [[linear combination]] of basis functionals, with coefficients ("components") ''u<sub>i</sub>'',
| |
| :<math>\tilde{u} = \sum_{i=1}^n u_i \, \tilde{\omega}^i. </math>
| |
| Then, applying the functional <math>\tilde{u}</math> to a basis vector ''e<sub>j</sub>'' yields
| |
| :<math>\tilde{u}(\vec e_j) = \sum_{i=1}^n (u_i \, \tilde{\omega}^i) \vec e_j = \sum_i u_i (\tilde{\omega}^i (\vec e_j)) </math>
| |
| due to linearity of scalar multiples of functionals and pointwise linearity of sums of functionals. Then
| |
| :<math> \tilde{u}({\vec e}_j) = \sum_i u_i (\tilde{\omega}^i ({\vec e}_j)) = \sum_i u_i \delta^i {}_j = u_j </math>
| |
| that is
| |
| :<math>\tilde{u} (\vec e_j) = u_j.</math>
| |
| This last equation shows that an individual component of a linear functional can be extracted by applying the functional to a corresponding basis vector. | |
| | |
| === The dual basis and inner product ===
| |
| When the space ''V'' carries an [[inner product]], then it is possible to write explicitly a formula for the dual basis of a given basis. Let ''V'' have (not necessarily orthogonal) basis <math>\vec{e}_1,\dots, \vec{e}_n</math>. In three dimensions (''n'' = 3), the dual basis can be written explicitly
| |
| :<math> \tilde{\omega}^i(\vec{v}) = {1 \over 2} \, \left\langle { \sum_{j=1}^3\sum_{k=1}^3\varepsilon^{ijk} \, (\vec e_j \times \vec e_k) \over \vec e_1 \cdot \vec e_2 \times \vec e_3} , \vec{v} \right\rangle.</math>
| |
| for ''i'' = 1, 2, 3, where ε is the [[Levi-Civita symbol]] and <math>\langle,\rangle</math> the inner product (or [[dot product]]) on ''V''.
| |
| | |
| In higher dimensions, this generalizes as follows
| |
| :<math> \tilde{\omega}^i(\vec{v}) = \left\langle \frac{\underset{{}^{1\le i_2<i_3<\dots<i_n\le n}}{\sum}\varepsilon^{ii_2\dots i_n}(\star \vec{e}_{i_2}\wedge\dots\wedge\vec{e}_{i_n})}{\star(\vec{e}_1\wedge\dots\wedge\vec{e}_n)}, \vec{v} \right\rangle </math>
| |
| where <math>\star</math> is the [[Hodge star operator]].
| |
| | |
| ==See also==
| |
| *[[Discontinuous linear map]]
| |
| *[[Positive linear functional]]
| |
| *[[Bilinear form]]
| |
| | |
| ==References==
| |
| *{{citation|first1=Richard|last1=Bishop|first2=Samuel|last2=Goldberg|year=1980|title=Tensor Analysis on Manifolds|publisher=Dover Publications|chapter=Chapter 4|isbn=0-486-64039-6}}
| |
| * {{citation|first=Paul|last=Halmos|authorlink=Paul Halmos|title=Finite dimensional vector spaces|year=1974|publisher=Springer|isbn=0-387-90093-4}}
| |
| * {{citation|authorlink=Peter Lax|first=Peter|last=Lax|title=Linear algebra|year=1996|publisher=Wiley-Interscience|isbn=978-0-471-11111-5}}
| |
| * {{Citation|first=Charles W. | last=Misner | first2=Kip. S. |last2=Thorne | first3=John A. | last3=Wheeler |title=Gravitation | publisher= W. H. Freeman | year=1973 | isbn=0-7167-0344-0 }}
| |
| *{{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Functional Analysis | publisher=McGraw-Hill Science/Engineering/Math | isbn=978-0-07-054236-5 | year=1991}}
| |
| * {{citation|first=Bernard|last=Schutz|year=1985|title=A first course in general relativity|publisher=Cambridge University Press|location=Cambridge, UK|chapter=Chapter 3|isbn=0-521-27703-5}}
| |
| | |
| {{reflist}}
| |
| | |
| {{Functional Analysis}}
| |
| | |
| [[Category:Functional analysis]]
| |
| [[Category:Linear algebra]]
| |
| [[Category:Linear operators]]
| |
Perfect lifetime is 1 which is free from any physical anguish. A individual experiencing pain caused by hemorrhoids could yearn for several treatment for hemorrhoids to ease his pain. However, nothing is more uplifting than to realize that you needn't undergo any kind of medicine or operation. In fact, there are a amount of signifies to avoid the appearance of hemorrhoids plus these preventive procedures are awfully simple.
The first symptoms of hemorrhoids is to utilize creams plus ointments. These lotions plus ointments is used found on the outer rectal region inside order to aid relaxing blood vessels. This can lower the inflammation because creams plus ointments will relax the tissue. But, this kind of treatment is considered to be wise for helping in really a short period. It is fairly possible that the hemorrhoid will probably to happen again.
There are many treatments that could be utilized for hemorrhoid. The first and the most popular is the cream and ointment. These are to be rubbed onto the affected piece of the anus. It helps to soothe the already inflamed blood vessels plus a momentary relief is accomplished. There is a relaxation of the tissues of the rectal column thus far the hemorrhoid is not thus much bulged. If there is a bulge yet, the pain relief may not do so much to aid.
Having a advantageous bowel movement etiquette is without doubt one of the right methods to avoid hemorrhoids. Everytime you feel the urge, do thus without hindrance. Numerous delays of going to the rest space brings about constipation that in turn causes you to strain difficult inside order to eliminate a bowels. As rapidly because you're finished, immediately receive up. Staying too long found on the bathroom seat may amplify pressure on your veins. After the bowel movement, you may moreover try sitting inside a tub with lukewarm water for 10 to 15 minutes to relax the rectum.
Right now, there are a great deal of hemorrhoid treatments. And yes, there are the painless hemorrhoid treatments furthermore available. Examples of such as use of petroleum jelly, the utilization of ointment phenylephrine or Preparation H, and even the easy utilize of soft cotton underwear. These are typically painless for with them we don't should go under the knife.
One of the number one methods of reducing yourself within the pain and itchiness is to soak in a warm Sitz bath. Do this for regarding 10 to 15 minutes, many instances a day. Warm water relaxes the muscles and reduces the swelling. You are able to additionally apply coconut oil to the affected region. It usually have a soothing impact though it won't last which lengthy. Applying witch hazel could equally aid soothe the pain plus itch. It decreases the bleeding and minimizes the swelling.
Whenever this shower is performed, we may then like to take some ice (wrapped up in a cloth) plus apply it to a anal region. This causes a decrease in the amount of blood which is flowing to the area; also ice may have a numbing effect giving we a small more relief from pain.