Revision #568 → #1290 · back to history
addedReal inner product842b0017b897
addedReal inner product space4e6cde1fa208
addedFinite-dimensional inner product spaces are Hilbert0eab229de5dd
addedAbsolutely convergent series of vectorse117cf558004
addedCompleteness of Euclidean space6bb0f3a6edd5
addedComplex modulus6198a0fdc74b
addedInner product on complex numbers2859b2a6fc6a
addedInner product on C²3624708592f9
addedHermitian symmetry of the C² inner product7e7c7fb92ab4
addedHilbert space6f3fe05d9457
addedComplex inner product space1884d254dee0
addedConjugate symmetry of inner product9c7d1a0fd6a9
addedLinearity in first argument6537313f43a7
addedPositive definiteness of inner productdfa8d71bb93b
addedAntilinearity in second argument61d70aa191c0
addedReal inner product spacec21e168c925c
addedNorm and distance from inner product41ba29ea7504
addedCauchy–Schwarz inequality860d60ade728
addedPre-Hilbert spaced26e925a2d91
addedHilbert space as complete pre-Hilbertdf6eb39a9542
addedCompleteness via Cauchy criterion / absolute convergence2fa84cfd2022
addedHilbert spaces are Banach spaces9a54c44824f4
addedClosed linear subspace is a Hilbert space6ac602c480a5
addedSequence space ℓ²7e3cff42149e
addedInner product on ℓ²be89102f359b
addedCompleteness of ℓ²70f1f4bb98b8
addedLebesgue space L²(X, μ)c8468bd6e9cf
addedInner product on L²b6b3408fb8f5
addedCompleteness of L²131473e830b5
addedWeighted L² space278b55b3f1ac
addedSobolev space H^s0ae6b75ff169
addedSobolev space as Bessel potentials551c1af565d0
addedHardy space H²(U)63a14693f536
addedHardy space via Fourier coefficientsa86194c564e7
addedBergman spaceb210a656dce1
addedBergman kernel reproducing property086cb2cc967b
addedReproducing kernel Hilbert spacea0e4e993e716
addedSturm–Liouville problem5a2f5a9d6014
addedLax–Milgram theorem7dbcdf1fc52e
addedPoisson equation weak formulationab7fdee7b315
addedErgodic dynamical system94d109c06910
addedVon Neumann mean ergodic theorem0e3f82f52b1d
addedTime average equals expectation for ergodic systemsa9088d6ab17b
addedOrthonormal basis of exponentials in L²([0,1])ceb46c191308
addedBest L² approximation via orthogonal projection411340eb32ff
addedPlancherel theorem66c4209b7514
addedQuantum mechanical state space formulationc18bacb00afe
addedPythagorean theorem of statisticsc63b731dd560
addedMartingale in a Hilbert spacebb23ab763db7
addedItô isometry8788cafe723e
addedOrthogonal vectors3ed46632de57
addedPythagorean identity in Hilbert spacec5ce707c9138
addedParallelogram identityae6f3209eb4b
addedPolarization identity characterization of Hilbert spaces58c43dbbf84a
addedHilbert spaces are uniformly convexe2b47d2e9c57
addedHilbert projection theorem438c2f2fda77
addedCharacterization of closest point in closed subspaced7d109d87b64
addedDensity criterion via orthogonal complement2f10ce116267
addedDual space of a Hilbert spacecbdb7646f2cc
addedRiesz representation theoremb3640cff4c60
addedHilbert spaces are reflexive8364c9be2a37
addedWeak convergence in Hilbert space597bd61953a8
addedAlaoglu's theorem (weak compactness of bounded sets)71a752d7ff30
addedMinimization of convex coercive functionals41e4cd0903c8
addedOpen mapping theorem2c425b4a06a3
addedBounded inverse theorem5151c6af510a
addedClosed graph theoremcb0f8e493ca4
addedGeometrical Hahn–Banach separation theorem55add34d6ada
addedBounded linear operator and operator normd16e16772f00
addedAdjoint of a bounded operatorab3735942378
addedB(H) is a C*-algebraeeb2575a9495
addedSelf-adjoint and positive operators1e0d00f727f1
addedSquare root of a positive operator7c3175e69594
addedSpectral theorem for bounded self-adjoint operatorsb0f3c73929e1
addedUnitary operatorc6ecc562ae52
addedCompact operator4616d56746f0
addedFredholm operator and indexdc526f40b271
addedUnbounded operatord761afb7cd91
addedDensely defined operatordb45d98dfc53
addedClosed and closable operatorsff2922440cbd
addedAdjoint of an unbounded operatorc158d48fd945
addedSymmetric and self-adjoint operatorsb6b57430e49d
addedPosition and momentum operators on L²(R)a42d51c75efb
addedDirect sum of Hilbert spacescf8221a2fcfa
addedInternal direct sum of closed subspacesb60491db1710
addedPeter–Weyl theoremae25ff7a5c95
addedHilbertian tensor product4e2de7911f20
addedTensor product of L² with itselfa4104d931a05
addedTensor product identifies with Hilbert–Schmidt operators08a6a56b9ecc
addedOrthonormal basis of a Hilbert space5ece704fd771
addedOrthonormal systeme137075b12ae
addedOrthonormal basis vs Hamel basiscd082a70ddf2
addedCompleteness criterion for orthonormal systemsa3a33727faea
addedStandard orthonormal bases39a96e784dec
addedSquare-summable sequence space ℓ²7d6f7186c801
addedUnit ball in ℓ² is not compactcd7aae598304
addedGeneralized ℓ²(B)2e0d05342b24
addedBessel's inequalitya7824d49cfb9
addedFourier expansion and Fourier coefficients9a037cb92d64
addedParseval's identity9d26734ed068
addedExistence of orthonormal basis (Hilbert dimension)4f185327e52b
addedIsometric isomorphism with ℓ²(B)ddb2f2857a4f
addedSeparable Hilbert spaced08ce02dd729
addedAll infinite-dimensional separable Hilbert spaces are isomorphic to ℓ²eefe2d72a733
addedOrthogonal complementfd884b07b8dc
addedDecomposition H = V ⊕ V⊥6641f63eb517
addedOrthogonal projection onto a closed subspaced839ea8e2251
addedMutually orthogonal projections843b80499e28
addedCharacterization of Hilbert spaces by projectionse30b28bdb0c5
addedTopological characterization via complementary subspacesfb2b8f846b98
addedClosure equals double orthogonal complement5d344c39c91b
addedOrthogonal projection theorem (box)05164cc257ad
addedSpectrum of an operatorf58adaecf4f5
addedSpectrum of self-adjoint operators is real and bounded25ffc151d90a
addedSpectral theorem for compact self-adjoint operators804de3f23f67
addedSpectral theorem for bounded self-adjoint operators via spectral family4e3c7e3f16d0
addedSpectral mapping theorem8b98fb79c7d6
addedSpectral theorem for unbounded self-adjoint operators (box)6c20e7031758