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  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.\nFor sequences $x = (x_n)$ and $y = (y_n)$ in $H$ with $\\sum_n \\|x_n\\|^2 \u003c \\infty$ and $\\sum_n \\|y_n\\|^2 \u003c \\infty$, let\n$$\\omega_{x,y}(T) = \\sum_{n} \\langle T x_n, y_n \\rangle, \\quad T \\in B(H).$$\nThe *ultraweak topology*, or *$\\sigma$-weak topology*, on $B(H)$ is the [[topology generated by the seminorms|ZVFXQ7S6FY]] $T \\mapsto \\abs{\\omega_{x,y}(T)}$ for all such pairs of sequences.\n\n## Remarks\n\nThe series defining $\\omega_{x,y}(T)$ converges absolutely, by the Cauchy–Schwarz inequality.\nEquivalently, the ultraweak topology is the weak-$*$ topology of the duality $B(H) \\cong \\mathcal{L}^1(H)^*$, where $\\mathcal{L}^1(H)$ is the space of [[trace class operators|Z2CP530WDE]] and $T$ acts by $\\rho \\mapsto \\operatorname{tr}(T \\rho)$.\n\nExample: for a trace class operator $\\rho$, the functional $T \\mapsto \\operatorname{tr}(T\\rho)$ is ultraweakly continuous.\nIndeed, writing $\\rho = \\sum_n s_n \\langle \\cdot, e_n \\rangle f_n$ in Schmidt form with $\\sum_n s_n \u003c \\infty$ gives $\\operatorname{tr}(T\\rho) = \\omega_{x,y}(T)$ with $x_n = \\sqrt{s_n} f_n$ and $y_n = \\sqrt{s_n} e_n$.\n\nNon-example: let $H$ be infinite-dimensional and let $\\varphi$ be a state on $B(H)$ that vanishes on all compact operators.\nSuch states exist because the quotient algebra by the compact operators has states.\nThen $\\varphi$ is norm-continuous but not ultraweakly continuous: every ultraweakly continuous functional has the form $T \\mapsto \\operatorname{tr}(T\\rho)$, vanishing on all rank-one operators forces $\\rho = 0$, and yet $\\varphi(I) = 1$.\n\nEvery [[weak operator topology|4XM1VN1C1N]] open set is ultraweakly open, and the two topologies agree on every norm-bounded subset of $B(H)$.\nThey differ on all of $B(H)$ when $H$ is infinite-dimensional.\n",
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