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cal, legal, financial, safety, and other critical issues? 8 Who owns Wikipedia? 9 Why am I having trouble logging in? 10 How can I contact Wikipedia? How do I create a new page? <span>You are required to have a Wikipedia account to create a new article—you can register here. To see other benefits to creating an account, see Why create an account? For creating a new article see Wikipedia:Your first article and Wikipedia:Article development; and you may wi

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A sunk cost is one that has already been incurred. You cannot change a sunk cost. Today’s decisions, on the other hand, should be based on current and future cash flows and should not be affected by prior, or sunk, costs. <span>An opportunity cost is what a resource is worth in its next-best use. For example, if a company uses some idle property, what should it record as the investment outlay: the purchase price several years ago, the current market value, or nothing? If you replace an old machine with a new one, what is the opportunity cost? If you invest $10 million, what is the opportunity cost? The answers to these three questions are, respective

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LAW 3 CONCEAL YOUR INTENTIONS JUDGMENT Keep people off-balance and in the dark by never revealing the purpose behind your actions. If they have no clue what you are up to, they cannot prepare a defense. Guide them f

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“IF YOU WANT TO GATHER HONEY, DON’T KICK OVER THE BEEHIVE”

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Dual space - Wikipedia Dual space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, ther

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In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

Dual space - Wikipedia Dual space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, ther

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f generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. <span>At a deeper level, perpendicular projection onto a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the

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mization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. <span>When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are sim

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nd only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. <span>Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between t

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of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) <span>In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0

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2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. <span>This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z ⟩ = ⟨ z , w

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also play a fundamental role in the foundations of quantum mechanics, being the natural space to hold all the possible states of a quantum system (possibly after projectivising the Hilbert space), but we will not discuss this subject here.) <span>Hilbert spaces are the natural abstract framework in which to study two important (and closely related) concepts: orthogonality and unitarity, allowing us to generalise familiar concepts and facts from Euclidean geometry such as the Cartesian coordinate system, rotations and reflections, and the Pythagorean theorem to Hilbert spaces. (For instance, the Fourier transform is a unitary transformation and can thus be viewed as a kind of generalised rotation.) Furthermore, the Hodge duality on Euclidean spaces has a partial analogue for Hilbert spaces, namely the Riesz representation theorem for Hilbert spaces, which makes the theory of duality and adjoints for Hilbert spaces especially simple (when compared with the more subtle theory of duality for, say, Banach spaces). Much later (next quarter, in fact), we will see that this duality allows us to extend the spectral theorem for self-adjoint matrices to that of self-adjoint operators on a Hilbert space. These notes are only the most basic introduction to the theory of Hilbert spaces. In particular, the theory of linear transformations between two Hilbert spaces, which is perhaps the

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al space) 3.2 R n (n-dimensional Euclidean space) 3.3 L 2 4 Applications 4.1 Analysis 4.2 Geometry 4.3 Probability theory 5 Generalizations 6 See also 7 Notes 8 References 9 External links Statement of the inequality[edit source] <span>The Cauchy–Schwarz inequality states that for all vectors u {\displaystyle u} and v {\displaystyle v} of an inner product space it is true that | ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product. Examples of inner products include the real and complex dot product, see the examples in inner product. Equivalently, by taking the square root of both sides, and referring to the norms

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) ≤ d ( x , y ) + d ( y , z ) . {\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.} [imagelink] <span>This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts | ⟨ x , y ⟩ |

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a, the free encyclopedia Jump to: navigation, search For usage in evolutionary biology, see Sequence space (evolution). For mathematical operations on sequence numbers, see Serial number arithmetic. <span>In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identif

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x i {\displaystyle \sum _{i=1}^{n}x_{i}} is not a norm because it may yield negative results. p-norm[edit source] Main article: L p space <span>Let p ≥ 1 be a real number. The p {\displaystyle p} -norm (also called ℓ p {\displaystyle \ell _{p}} -norm) of vectors x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is ‖ x ‖ p := ( ∑ i = 1 n | x i | p ) 1 / p . {\displaystyle \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}.} For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches ∞ {\displaystyle \infty } the p-norm approa

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Let p ≥ 1 be a real number. The -norm of vectors is

x i {\displaystyle \sum _{i=1}^{n}x_{i}} is not a norm because it may yield negative results. p-norm[edit source] Main article: L p space <span>Let p ≥ 1 be a real number. The p {\displaystyle p} -norm (also called ℓ p {\displaystyle \ell _{p}} -norm) of vectors x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is ‖ x ‖ p := ( ∑ i = 1 n | x i | p ) 1 / p . {\displaystyle \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}.} For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches ∞ {\displaystyle \infty } the p-norm approa

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Let p ≥ 1 be a real number. The -norm of vectors is

x i {\displaystyle \sum _{i=1}^{n}x_{i}} is not a norm because it may yield negative results. p-norm[edit source] Main article: L p space <span>Let p ≥ 1 be a real number. The p {\displaystyle p} -norm (also called ℓ p {\displaystyle \ell _{p}} -norm) of vectors x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is ‖ x ‖ p := ( ∑ i = 1 n | x i | p ) 1 / p . {\displaystyle \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}.} For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches ∞ {\displaystyle \infty } the p-norm approa

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Lp space - Wikipedia L p space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). L p

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In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces,

Lp space - Wikipedia L p space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). L p

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In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces,

Lp space - Wikipedia L p space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). L p

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In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.

Lp space - Wikipedia L p space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). L p

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In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.

Lp space - Wikipedia L p space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). L p

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In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers

a, the free encyclopedia Jump to: navigation, search For usage in evolutionary biology, see Sequence space (evolution). For mathematical operations on sequence numbers, see Serial number arithmetic. <span>In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identif

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In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers

a, the free encyclopedia Jump to: navigation, search For usage in evolutionary biology, see Sequence space (evolution). For mathematical operations on sequence numbers, see Serial number arithmetic. <span>In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identif

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ected from Eigenfunction expansion) Jump to: navigation, search [imagelink] This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk. <span>In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ. [1] [2] [3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvect

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In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that for some scalar eigenvalue λ.

ected from Eigenfunction expansion) Jump to: navigation, search [imagelink] This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk. <span>In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ. [1] [2] [3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvect

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In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that for some scalar eigenvalue λ.

ected from Eigenfunction expansion) Jump to: navigation, search [imagelink] This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk. <span>In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ. [1] [2] [3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvect

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a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. <span>The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts. Augustin-Louis Cauchy proved the spectral theorem for self-adjoint matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systema

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The spectral theorem provides a spectral decomposition of the underlying vector space on which the operator acts.

a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. <span>The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts. Augustin-Louis Cauchy proved the spectral theorem for self-adjoint matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systema

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Integral equation From Wikipedia, the free encyclopedia (Redirected from Integral equations) Jump to: navigation, search For equations of integer unknowns, see Diophantine equation. <span>In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm theory, and Maxwe

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rview 2 Numerical solution 3 Classification 4 Wiener–Hopf integral equations 5 Power series solution for integral equations 6 Integral equations as a generalization of eigenvalue equations 7 See also 8 References 9 External links Overview<span>[edit source] The most basic type of integral equation is called a Fredholm equation of the first type, f ( x ) = ∫ a b K ( x , t ) φ ( t ) d t . {\displaystyle f(x)=\int _{a}^{b}K(x,t)\,\varphi (t)\,dt.} The notation follows Arfken. Here φ is an unknown function, f is a known function, and K is another known function of two variables, often called the kernel function. Note that the limits of integration are constant: this is what characterizes a Fredholm equation. If the unknown function occurs both inside and outside of the integral, the equation

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In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.

Integral equation From Wikipedia, the free encyclopedia (Redirected from Integral equations) Jump to: navigation, search For equations of integer unknowns, see Diophantine equation. <span>In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm theory, and Maxwe

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In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.

Integral equation From Wikipedia, the free encyclopedia (Redirected from Integral equations) Jump to: navigation, search For equations of integer unknowns, see Diophantine equation. <span>In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm theory, and Maxwe

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The most basic type of integral equation is where φ is an unknown function, f is a known function, and K is another known function of two variables, often called the kernel function.

rview 2 Numerical solution 3 Classification 4 Wiener–Hopf integral equations 5 Power series solution for integral equations 6 Integral equations as a generalization of eigenvalue equations 7 See also 8 References 9 External links Overview<span>[edit source] The most basic type of integral equation is called a Fredholm equation of the first type, f ( x ) = ∫ a b K ( x , t ) φ ( t ) d t . {\displaystyle f(x)=\int _{a}^{b}K(x,t)\,\varphi (t)\,dt.} The notation follows Arfken. Here φ is an unknown function, f is a known function, and K is another known function of two variables, often called the kernel function. Note that the limits of integration are constant: this is what characterizes a Fredholm equation. If the unknown function occurs both inside and outside of the integral, the equation

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Square-integrable function - Wikipedia Square-integrable function From Wikipedia, the free encyclopedia (Redirected from Square-integrable) Jump to: navigation, search <span>In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, if ∫ − ∞ ∞ | f

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In mathematics, a square-integrable function is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

Square-integrable function - Wikipedia Square-integrable function From Wikipedia, the free encyclopedia (Redirected from Square-integrable) Jump to: navigation, search <span>In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, if ∫ − ∞ ∞ | f

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In mathematics, a square-integrable function is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

Square-integrable function - Wikipedia Square-integrable function From Wikipedia, the free encyclopedia (Redirected from Square-integrable) Jump to: navigation, search <span>In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, if ∫ − ∞ ∞ | f

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The Cauchy–Schwarz inequality states that for all vectors and of an inner product space it is true that

al space) 3.2 R n (n-dimensional Euclidean space) 3.3 L 2 4 Applications 4.1 Analysis 4.2 Geometry 4.3 Probability theory 5 Generalizations 6 See also 7 Notes 8 References 9 External links Statement of the inequality[edit source] <span>The Cauchy–Schwarz inequality states that for all vectors u {\displaystyle u} and v {\displaystyle v} of an inner product space it is true that | ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product. Examples of inner products include the real and complex dot product, see the examples in inner product. Equivalently, by taking the square root of both sides, and referring to the norms

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The Cauchy–Schwarz inequality states that for all vectors and of an inner product space it is true that

al space) 3.2 R n (n-dimensional Euclidean space) 3.3 L 2 4 Applications 4.1 Analysis 4.2 Geometry 4.3 Probability theory 5 Generalizations 6 See also 7 Notes 8 References 9 External links Statement of the inequality[edit source] <span>The Cauchy–Schwarz inequality states that for all vectors u {\displaystyle u} and v {\displaystyle v} of an inner product space it is true that | ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product. Examples of inner products include the real and complex dot product, see the examples in inner product. Equivalently, by taking the square root of both sides, and referring to the norms

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This inner product in the space ℂ 2 is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate:

2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. <span>This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z ⟩ = ⟨ z , w

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In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite.

of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) <span>In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0

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In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite.

of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) <span>In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0

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Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.

is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. <span>Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum. Contents [hide] 1 Definition and illustration 1.1 Motivating example: Euclidean space 1.2 Definition 1.3 Second example: sequence spaces 2 History 3 Examples 3.1 Lebesgu

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Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.

is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. <span>Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum. Contents [hide] 1 Definition and illustration 1.1 Motivating example: Euclidean space 1.2 Definition 1.3 Second example: sequence spaces 2 History 3 Examples 3.1 Lebesgu

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When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable.

mization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. <span>When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are sim

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When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable.

mization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. <span>When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are sim

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ombates UFC Deportes Aventura eSports Juegos Olímpicos Olimpismo Paralímpicos Rugby Toros Turf Volvo Ocean Race Ligue 1 - Francia Liga FrancesaIsaac se formó en el Chelsea y ahora ficha por otro exequipo de Didier <span>El hijo de Drogba ficha por el Guingamp siguiendo los pasos de su padre Compartir en Facebook Compartir en Twitter Enviar por email 20/02/2018 16:06 CET Isaac Drogba, durante su presentación con el Guingamp. 3 comentarios Comentar Guardiola vuelve a sa

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El hijo de Drogba ficha por el Guingamp siguiendo los pasos de su padre

ombates UFC Deportes Aventura eSports Juegos Olímpicos Olimpismo Paralímpicos Rugby Toros Turf Volvo Ocean Race Ligue 1 - Francia Liga FrancesaIsaac se formó en el Chelsea y ahora ficha por otro exequipo de Didier <span>El hijo de Drogba ficha por el Guingamp siguiendo los pasos de su padre Compartir en Facebook Compartir en Twitter Enviar por email 20/02/2018 16:06 CET Isaac Drogba, durante su presentación con el Guingamp. 3 comentarios Comentar Guardiola vuelve a sa

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At a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory.

f generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. <span>At a deeper level, perpendicular projection onto a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the

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