Let $1\leq p<\infty$. Are the Banach spaces $L^p(\mathbb{R})$ and $L^p[0,1]$ isomorphic?
The case $p=\infty$ can be treated by seeing that $\mathbb{R}$ and $(0,1)$ are homeomorphic.
Can this be extended to $0<p<1$?
2026-03-27 14:51:32.1774623092
Can $L^p(\mathbb{R})$ and $L^p[0,1]$ be isomorphic?
385 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in MEASURE-THEORY
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Absolutely continuous functions are dense in $L^1$
- I can't undestand why $ \{x \in X : f(x) > g(x) \} = \bigcup_{r \in \mathbb{Q}}{\{x\in X : f(x) > r\}\cap\{x\in X:g(x) < r\}} $
- Trace $\sigma$-algebra of a product $\sigma$-algebra is product $\sigma$-algebra of the trace $\sigma$-algebras
- Meaning of a double integral
- Random variables coincide
- Convergence in measure preserves measurability
- Convergence in distribution of a discretized random variable and generated sigma-algebras
- A sequence of absolutely continuous functions whose derivatives converge to $0$ a.e
- $f\in L_{p_1}\cap L_{p_2}$ implies $f\in L_{p}$ for all $p\in (p_1,p_2)$
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 LP-SPACES
- Absolutely continuous functions are dense in $L^1$
- Understanding the essential range
- Problem 1.70 of Megginson's "An Introduction to Banach Space Theory"
- Showing a sequence is in $\ell^1$
- How to conclude that $\ell_\infty$ is not separable from this exercise?
- Calculating the gradient in $L^p$ space when $0<p<1$ and the uderlying set is discrete and finite
- $f_{n} \in L^{p}(X),$ such that $\lVert f_{n}-f_{n+1}\rVert_{p} \leq \frac{1}{n^2}$. Prove $f_{n}$ converges a.e.
- Find a sequence converging in distribution but not weakly
- Elementary use of Hölder inequality
- Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$
Related Questions in VECTOR-SPACE-ISOMORPHISM
- Showing that $ \text{Ind}_H^G W \cong \text{Ind}_K^G(\text{Ind}_H^K W)$
- if $T$ is isomorphism, how can I prove that $[T^{-1}]_B=[T]_B^{-1}$ for any base $B$ of $V$?
- Proofs on Isomorphism Problems
- Basis of vector spaces in perfect pairing
- Linear isomorphism of quotient spaces
- $V$ and $\mathcal{L}(\mathbf{F},V)$ are isomorphic
- Isomorphic Hilbert spaces iff they have the same dimension
- Vector space isomorphic to direct sum
- Trying to find the dimension of a vector space...
- $V^*$ is isomorphic to the direct product of copies of $F$ indexed by $A$
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?
Hint. Your isomorphism $T : L^p(\mathbb R) \to L^p(0,1)$ will have the form $T(f)(x) = \varphi(x) f(\psi(x))$ for appropriate functions $\varphi, \psi$.