In class, we proved that weak operator and ultraweak topologies coincide on norm-bounded subsets of $\mathcal{B} (\mathcal{H})$ (here, $\mathcal{H}$ is a Hilbert space) using Banach-Alaoglu. This had me wondering whether the same is also true for strong operator and ultrastrong topologies. Wikipedia claims it is, but doesn't give a reference. Does there exist an elementary proof of this?
2026-04-24 08:11:42.1777018302
Strong operator and ultrastrong topologies are the same on bounded subsets
33 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in FUNCTIONAL-ANALYSIS
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- Prove or disprove the following inequality
- Unbounded linear operator, projection from graph not open
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- Elementary question on continuity and locally square integrability of a function
- Bijection between $\Delta(A)$ and $\mathrm{Max}(A)$
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- If $A$ generates the $C_0$-semigroup $\{T_t;t\ge0\}$, then $Au=f \Rightarrow u=-\int_0^\infty T_t f dt$?
Related Questions in OPERATOR-THEORY
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- Confusion about relationship between operator $K$-theory and topological $K$-theory
- Definition of matrix valued smooth function
- hyponormal operators
- a positive matrix of operators
- If $S=(S_1,S_2)$ hyponormal, why $S_1$ and $S_2$ are hyponormal?
- Closed kernel of a operator.
- Why is $\lambda\mapsto(\lambda\textbf{1}-T)^{-1}$ analytic on $\rho(T)$?
- Show that a sequence of operators converges strongly to $I$ but not by norm.
- Is the dot product a symmetric or anti-symmetric operator?
Related Questions in HILBERT-SPACES
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- hyponormal operators
- a positive matrix of operators
- If $S=(S_1,S_2)$ hyponormal, why $S_1$ and $S_2$ are hyponormal?
- Is the cartesian product of two Hilbert spaces a Hilbert space?
- Show that $ Tf $ is continuous and measurable on a Hilbert space $H=L_2((0,\infty))$
- Kernel functions for vectors in discrete spaces
- The space $D(A^\infty)$
- Show that $Tf$ is well-defined and is continious
- construction of a sequence in a complex Hilbert space which fulfills some specific properties
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?