Revision #1840 → #2351 · back to history
modifiedEuclidean normee23051f2837
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"The p -norm in finite dimensions","snippet":"The Euclidean length of a vector"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{2}=\\left({x_{1}}^{2}+{x_{2}}^{2}+\\dotsb +{x_{n}}^{2}\\right)^{1/2}.}"}] | — |
modifiedp-norm in finite dimensions325e1575c0b8
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"The p -norm in finite dimensions","snippet":"For a real number"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{p}=\\left(|x_{1}|^{p}+|x_{2}|^{p}+\\dotsb +|x_{n}|^{p}\\right)^{1/p}.}"}] | — |
modifiedMaximum norm (infinity-norm)fb6459086c7c
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"The p -norm in finite dimensions","snippet":"maximum norm (or uniform norm) is the limit"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{\\infty }=\\max \\left\\{|x_{1}|,|x_{2}|,\\dotsc ,|x_{n}|\\right\\}}"}] | — |
modifiedEuclidean norm bounded by 1-normf19eeb645f43
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Relations between p -norms","snippet":"the Euclidean norm of any vector is bounded by its 1-norm"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{2}\\leq \\|x\\|_{1}.}"}] | — |
modifiedL^0 norm (Banach F-norm)b6779ed2088f
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"When p = 0","snippet":"The mathematical definition of the"},{"type":"math_alttext","value":"{\\displaystyle (x_{n})\\mapsto \\|x\\|:=d(0,x)=\\sum _{n}2^{-n}{\\frac {|x_{n}|}{1+|x_{n}|}}.}"}] | — |
modifiedZero "norm" (Donoho)4ac134c57f61
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"When p = 0","snippet":"is the number of non-zero entries of the vector"},{"type":"math_alttext","value":"{\\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\\cdots +|x_{n}|^{0}.}"}] | — |
modifiedℓ^p-norm on sequences8a7c277b3255
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"ℓ p spaces and sequence spaces","snippet":"Define the"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{p}=\\left(|x_{1}|^{p}+|x_{2}|^{p}+\\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\\cdots \\right)^{1/p}}"}] | — |
modifiedHarmonic sequence in ℓ^pc3f913ebe2eb
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"ℓ p spaces and sequence spaces","snippet":"as the series"},{"type":"math_alttext","value":"{\\displaystyle \\left(1,{\\frac {1}{2}},\\ldots ,{\\frac {1}{n}},{\\frac {1}{n+1}},\\ldots \\right)}"},{"type":"math_alttext","value":"{\\displaystyle 1^{p}+{\\frac {1}{2^{p}}}+\\cdots +{\\frac {1}{n^{p}}}+{\\frac {1}{(n+1)^{p}}}+\\cdots ,}"}] | — |
modifiedℓ^∞ norm and space359c26437b81
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"ℓ p spaces and sequence spaces","snippet":"One also defines the"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{\\infty }=\\sup(|x_{1}|,|x_{2}|,\\dotsc ,|x_{n}|,|x_{n+1}|,\\ldots )}"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{\\infty }=\\lim _{p\\to \\infty }\\|x\\|_{p}}"}] | — |
modifiedGeneral ℓ^p(I) space4c51f6873dec
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"General ℓ p -space","snippet":"In complete analogy to the preceding definition"},{"type":"math_alttext","value":"{\\displaystyle \\ell ^{p}(I)=\\left\\{(x_{i})_{i\\in I}\\in \\mathbb {K} ^{I}:\\sum _{i\\in I}|x_{i}|^{p}<+\\infty \\right\\},}"},{"type":"math_alttext","value":"{\\displaystyle \\|x\\|_{p}=\\left(\\sum _{i\\in I}|x_{i}|^{p}\\right)^{1/p}}"}] | — |
modifiedL^p space (p finite)5756ebd7db6c
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"L p spaces and Lebesgue integrals","snippet":"consider the set"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{p}~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~\\left(\\int _{S}|f|^{p}\\;\\mathrm {d} \\mu \\right)^{1/p}<\\infty .}"}] | — |
modifiedEqual almost everywhere593212f8c33c
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"L p spaces and Lebesgue integrals","snippet":"are said to be equal almost everywhere"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{\\infty }~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~\\inf\\{C\\in \\mathbb {R} _{\\geq 0}:|f(s)|\\leq C{\\text{ for almost every }}s\\}.}"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{\\infty }~=~{\\begin{cases}\\operatorname {esssup} |f|&{\\text{if }}\\mu (S)>0,\\\\0&{\\text{if }}\\mu (S)=0.\\end{cases}}}"}] | — |
modifiedMinkowski inequality / triangle inequality for L^p972193f6c025
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"L p spaces and Lebesgue integrals","snippet":"it is also a consequence of Minkowski's inequality"},{"type":"math_alttext","value":"{\\displaystyle \\|f+g\\|_{p}\\leq \\|f\\|_{p}+\\|g\\|_{p}}"}] | — |
modifiedL^p-norm on quotientb3ae4436346a
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"L p spaces and Lebesgue integrals","snippet":"called the"},{"type":"math_alttext","value":"{\\displaystyle \\|f+{\\mathcal {N}}\\|_{p}\\;{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}\\;\\|f\\|_{p}.}"},{"type":"math_alttext","value":"{\\displaystyle L^{p}(S,\\mu )~~{\\stackrel {\\scriptscriptstyle {\\text{def}}}{=}}~~{\\mathcal {L}}^{p}(S,\\mu )/{\\mathcal {N}}~=~\\{f+{\\mathcal {N}}:f\\in {\\mathcal {L}}^{p}(S,\\mu )\\}.}"}] | — |
modifiedL^2 is the only Hilbert L^p96aceac03015
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Special cases","snippet":"is the only Hilbert space among"},{"type":"math_alttext","value":"{\\displaystyle \\langle f,g\\rangle =\\int _{S}f(x){\\overline {g(x)}}\\,\\mathrm {d} \\mu (x).}"}] | — |
modifiedInner product on L^26693fc807c0f
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Special cases","snippet":"the inner product on"},{"type":"math_alttext","value":"{\\displaystyle \\langle f,g\\rangle =\\int _{S}f(x){\\overline {g(x)}}\\,\\mathrm {d} \\mu (x).}"}] | — |
modifiedHölder's inequality21458d17eac2
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Hölder's inequality","snippet":"This inequality, called Hölder's inequality"},{"type":"math_alttext","value":"{\\displaystyle \\sup _{\\|g\\|_{q}\\leq 1}\\,\\int _{S}|fg|\\,\\mathrm {d} \\mu ~<~\\infty }"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{p}~=~\\sup _{\\|g\\|_{q}\\leq 1}\\,\\int _{S}fg\\,\\mathrm {d} \\mu .}"}] | — |
modifiedGeneralized Minkowski inequality5e0dc6937358
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Generalized Minkowski inequality","snippet":"Minkowski inequality , which states that"},{"type":"math_alttext","value":"{\\displaystyle \\left\\|\\left\\|F(\\,\\cdot ,n)\\right\\|_{L^{p}(M,\\mu )}\\right\\|_{L^{q}(N,\\nu )}~\\leq ~\\left\\|\\left\\|F(m,\\cdot )\\right\\|_{L^{q}(N,\\nu )}\\right\\|_{L^{p}(M,\\mu )}\\ .}"}] | — |
modifiedAtomic decomposition for L^p3273c04303fd
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Atomic decomposition","snippet":"has an atomic decomposition"},{"type":"math_alttext","value":"{\\displaystyle f~=~\\sum _{n\\in \\mathbb {Z} }r_{n}\\,f_{n}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle \\|f_{n}\\|_{\\infty }~\\leq ~2^{-{\\tfrac {n}{p}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\tfrac {1}{2}}\\|f\\|_{p}^{p}~\\leq ~\\sum _{n\\in \\mathbb {Z} }r_{n}^{p}~\\leq ~2\\|f\\|_{p}^{p}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{p}^{p}~=~\\sum _{n\\in \\mathbb {Z} }r_{n}^{p}\\,\\|f_{n}\\|_{p}^{p}\\,.}"}] | — |
modifiedExplicit construction of atomic decomposition536314592ed6
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Atomic decomposition","snippet":"An atomic decomposition can be explicitly given"},{"type":"math_alttext","value":"{\\displaystyle t_{n}=\\inf\\{t\\in \\mathbb {R} :\\mu (f>t)<2^{n}\\}}"},{"type":"math_alttext","value":"{\\displaystyle r_{n}~=~2^{n/p}\\,t_{n}~{\\text{ and }}\\quad f_{n}~=~{\\frac {f}{r_{n}}}\\,\\mathbf {1} _{(t_{n+1}<f\\leq t_{n})}}"}] | — |
modifiedLayer-cake formula for L^p-norm36cf5eb5ab36
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Atomic decomposition","snippet":"The complementary cumulative distribution function"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{p}^{p}~=~p\\,\\int _{0}^{\\infty }t^{p-1}\\mu (|f|>t)\\,\\mathrm {d} t\\,,}"}] | — |
modifiedDual of L^p is L^q59d56e25e684
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Dual spaces","snippet":"The dual space of"},{"type":"math_alttext","value":"{\\displaystyle f\\mapsto \\kappa _{p}(g)(f)=\\int fg\\,\\mathrm {d} \\mu }"}] | — |
modifiedContinuity of embedding via closed graph073c36840ef1
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Embeddings","snippet":"As a consequence of the closed graph theorem , the embedding is continuous"},{"type":"math_alttext","value":"{\\displaystyle \\ \\|\\mathbf {1} f^{p}\\|_{1}\\leq \\|\\mathbf {1} \\|_{q/(q-p)}\\|f^{p}\\|_{q/p}}"},{"type":"math_alttext","value":"{\\displaystyle \\ \\|f\\|_{p}\\leq \\mu (S)^{1/p-1/q}\\|f\\|_{q}.}"}] | — |
modifiedOptimal constant of embedding1059d4281131
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Embeddings","snippet":"The constant appearing in the above inequality is optimal"},{"type":"math_alttext","value":"{\\displaystyle \\|I\\|_{q,p}=\\mu (S)^{1/p-1/q}}"}] | — |
modifiedDensity of integrable simple functions8722df80874f
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Dense subspaces","snippet":"the vector space of integrable simple functions is dense"},{"type":"math_alttext","value":"{\\displaystyle f=\\sum _{j=1}^{n}a_{j}\\mathbf {1} _{A_{j}},}"}] | — |
modifiedUrysohn approximation in normal spacesc77d10da61eb
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Dense subspaces","snippet":"there exists a Urysohn function"},{"type":"math_alttext","value":"{\\displaystyle F\\subseteq A\\subseteq U\\subseteq V\\quad {\\text{and}}\\quad \\mu (U\\setminus F)=\\mu (U)-\\mu (F)<\\varepsilon ,}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{S}|\\mathbf {1} _{A}-\\varphi |\\,\\mathrm {d} \\mu <\\varepsilon \\,.}"}] | — |
modifiedDistribution functionce6d8c0a6a5d
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Weak L p","snippet":"The distribution function of"},{"type":"math_alttext","value":"{\\displaystyle \\lambda _{f}(t)=\\mu \\{x\\in S:|f(x)|>t\\}.}"}] | — |
modifiedMarkov's inequality bound45410e37db2d
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Weak L p","snippet":"then by Markov's inequality"},{"type":"math_alttext","value":"{\\displaystyle \\lambda _{f}(t)\\leq {\\frac {\\|f\\|_{p}^{p}}{t^{p}}}}"}] | — |
modifiedWeak L^p space5a61962c8482
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Weak L p","snippet":"is said to be in the space weak"},{"type":"math_alttext","value":"{\\displaystyle \\lambda _{f}(t)\\leq {\\frac {C^{p}}{t^{p}}}}"}] | — |
modifiedWeak L^p quasi-norm6e1b8cef45cc
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Weak L p","snippet":"The best constant"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{p,w}=\\sup _{t>0}~t\\lambda _{f}^{1/p}(t).}"}] | — |
modifiedWeak L^p quasi-triangle inequality89ec921e51df
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Weak L p","snippet":"is not a true norm, since the triangle inequality fails"},{"type":"math_alttext","value":"{\\displaystyle \\|f\\|_{p,w}\\leq \\|f\\|_{p}}"}] | — |
modifiedWeak L^p comparable to L^p norm; Banach for p>1e3b28eabe797
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"Weak L p","snippet":"is comparable to the"},{"type":"math_alttext","value":"{\\displaystyle \\||f|\\|_{L^{p,\\infty }}=\\sup _{0<\\mu (E)<\\infty }\\mu (E)^{-1/r+1/p}\\left(\\int _{E}|f|^{r}\\,d\\mu \\right)^{1/r}}"}] | — |
modifiedLévy metric on L^08d3603f3a17f
| Field | From #1840 | To #2351 |
|---|
| anchors | [{"section":"L 0 space of measurable functions","snippet":"Such a metric is called"},{"type":"math_alttext","value":"{\\displaystyle d(f,g)=\\int _{S}\\varphi {\\bigl (}|f(x)-g(x)|{\\bigr )}\\,\\mathrm {d} \\mu (x)}"}] | — |
addedPolarization identity on ℓ^2771b7978ee47
addedL^p is a Banach space (general)48efdaaea69b