I got a question concerning the definition of the $L^p$-spaces.
Is it true that $L^p[0,2π[=L^p[0,2π]$?
2026-03-30 20:50:36.1774903836
Lebesgue Space Definition
35 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in LEBESGUE-INTEGRAL
- A sequence of absolutely continuous functions whose derivatives converge to $0$ a.e
- Square Integrable Functions are Measurable?
- Lebesgue measure and limit of the integral.
- Solving an integral by using the Dominated Convergence Theorem.
- Convergence of a seqence under the integral sign
- If $g \in L^1$ and $f_n \to f$ a.e. where $|f_n| \leq 1$, then $g*f_n \to g*f$ uniformly on each compact set.
- Integral with Dirac measure.
- If $u \in \mathscr{L}^1(\lambda^n), v\in \mathscr{L}^\infty (\lambda^n)$, then $u \star v$ is bounded and continuous.
- Proof that $x \mapsto \int |u(x+y)-u(y)|^p \lambda^n(dy)$ is continuous
- a) Compute $T(1_{[\alpha,\beta]})$ for all $0<\alpha <\beta<0$
Related Questions in LEBESGUE-MEASURE
- A sequence of absolutely continuous functions whose derivatives converge to $0$ a.e
- property of Lebesgue measure involving small intervals
- Is $L^p(\Omega)$ separable over Lebesgue measure.
- Lebesgue measure and limit of the integral.
- uncountable families of measurable sets, in particular balls
- Joint CDF of $X, Y$ dependent on $X$
- Show that $ Tf $ is continuous and measurable on a Hilbert space $H=L_2((0,\infty))$
- True or False Question on Outer measure.
- Which of the following is an outer measure?
- Prove an assertion for a measure $\mu$ with $\mu (A+h)=\mu (A)$
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?
They are not the same set. One is made up by (equivalence classes of) functions with domain $[0,2\pi]$ and the other of functions with domain $[0,2\pi[$. Their elements are different (remember that a function is a triple: domain, codomain, and subset of their cartesian product), and thus they are different sets. However, they are 'morally' the same vector space: the function $\Psi: L^{p}[0,2\pi] \to L^{p}[0,2\pi[$ that takes $[f]$ to $[f \vert_{[0, 2\pi[}]$ is a linear isomorphism. Surjectivity follows since one can simply extend a function in the codomain to a function in the domain by adding the value $0$ (or any value, really) in $2\pi$.