What is the topology of the "character group"? Is it locally compact (and Hausdorff)? Which book say about them?
Is there a well known topology on the "character group" when the primer group is Abelian and locally compact (Hausdorff)?
2026-03-29 09:11:53.1774775513
What is the topology of the "character group"?
221 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in REPRESENTATION-THEORY
- How does $\operatorname{Ind}^G_H$ behave with respect to $\bigoplus$?
- Minimal dimension needed for linearization of group action
- How do you prove that category of representations of $G_m$ is equivalent to the category of finite dimensional graded vector spaces?
- Assuming unitarity of arbitrary representations in proof of Schur's lemma
- Are representation isomorphisms of permutation representations necessarily permutation matrices?
- idempotent in quiver theory
- Help with a definition in Serre's Linear Representations of Finite Groups
- Are there special advantages in this representation of sl2?
- Properties of symmetric and alternating characters
- Representation theory of $S_3$
Related Questions in TOPOLOGICAL-GROUPS
- Are compact groups acting on Polish spaces essentially Polish?
- Homotopy group of rank 2 of various manifolds
- A question on Group of homeomorphism of $[0,1]$.
- $G\cong G/H\times H$ measurably
- Is a connected component a group?
- How to realize the character group as a Lie/algebraic/topological group?
- Show $\widehat{\mathbb{Z}}$ is isomorphic to $S^1$
- a question on Ellis semigroup
- Pontryagin dual group inherits local compactness
- Property of the additive group of reals
Related Questions in CHARACTERS
- Show that for character $\chi$ of an Abelian group $G$ we have $[\chi; \chi] \ge \chi(1)$.
- Properties of symmetric and alternating characters
- Counting characters of bounded conductors
- The condition between $\chi(1)$ and $[G:H]$ which gives us a normal subgroup.
- How to realize the character group as a Lie/algebraic/topological group?
- Show $\widehat{\mathbb{Z}}$ is isomorphic to $S^1$
- Confusion about the conductor for Dirichlet characters
- Relation between characters of symmetric group and general linear group
- Information on semilinear groups.
- Let $ f $ be an irreducible polynomial in $ \mathbb{F }_q [x] $, why $ f ^\frac{s}{deg (f)} $ has degree term $ s-1 $?
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?
The situation is thus: let $G$ be locally compact abelian and $\Gamma$ be its dual group i.e. its group of continuous characters. The Fourier transform takes functions in $L^1(G)$ to functions on $\Gamma$. If $A$ is the image of $L^1(G)$ under the Fourier transform, then the weak topology induced by $A$ on $\Gamma$ makes $\Gamma$ a locally compact Hausdorff space, and $\Gamma$ is a locally compact abelian topological group. A good reference is Rudin's Fourier Analysis on Groups.