If $(X, \|\cdot\|_{1})$ is a reflexive Banach space and $\|\cdot\|_{2}$ is a norm on $X$ such that $\|\cdot\|_{2} \leq \|\cdot\|_{1}$, is $(X, \|\cdot\|_{2})$ also reflexive?
2026-02-23 08:22:52.1771834972
Reflexivity carried to the lesser norm
27 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in NORMED-SPACES
- How to prove the following equality with matrix norm?
- Closure and Subsets of Normed Vector Spaces
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Minimum of the 2-norm
- Show that $\Phi$ is a contraction with a maximum norm.
- Understanding the essential range
- Mean value theorem for functions from $\mathbb R^n \to \mathbb R^n$
- Metric on a linear space is induced by norm if and only if the metric is homogeneous and translation invariant
- Gradient of integral of vector norm
Related Questions in BANACH-SPACES
- Problem 1.70 of Megginson's "An Introduction to Banach Space Theory"
- Is the cartesian product of two Hilbert spaces a Hilbert space?
- Why is $\lambda\mapsto(\lambda\textbf{1}-T)^{-1}$ analytic on $\rho(T)$?
- Is ${C}[0,1],\Bbb{R}$ homeomorphic to any $\Bbb{R^n}$, for an integer $n$?
- Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$
- Theorem 1.7.9 of Megginson: Completeness is a three-space property.
- A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded.
- Separability of differentiable functions
- Showing $u_{\lambda}(x):= \left(\frac{\lambda}{{\lambda}^{2}+|x|^2}\right)^{\frac{n-2}{2}}$ is not sequentially compact in $L^{2^{*}}$
- Proving that a composition of bounded operator and trace class operator is trace class
Related Questions in REFLEXIVE-SPACE
- Weak convergence in a reflexive, separable and infinite Banach space.
- Weak and Weak* convergences implying reflexivity
- Non-Reflexivity of $L^1$ by finding a particular operator in the dual space?
- My attempt to show the canonical embedding $c_0\rightarrow c_0^{**}$ is not surjective?
- Is the Unit ball of $X^{**} $ weak*- sequentially compact?
- How to get $\ell_1$ is not reflexive from $c_0^*$ is not reflexive?
- Example on $X^*$ is reflexive but $\sigma(X^*,X)\neq\sigma(X^*,X^{**})$ for a normed space $X$.
- A direct example on $X^*$ is reflexive but the weak topology and weak* topology on $X^*$ are not equal for a normed space $X$.
- How to show a normed space $X$ is reflexive if the weak topology and weak* topology on $X^*$ are equal?
- Bounded sequence has Cauchy subsequence w.r.t. $ (x|y)_0:=\sum^\infty_{n=1} 2^{-n}\phi_n(x)\phi_n(y).$
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?
By the Open Mapping Theorem, $(X,||\cdot||_1)$ and $(X,||\cdot||_2)$ are isomorphic, so $(X,||\cdot||_2)$ is reflexive too