# Questions tagged [lp-spaces]

For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a problem is a $L^p$ space.

3,278 questions
46 views

35 views

### Show {$e^{2\pi i bnx}$} is a basis for $L^2[0,b^{-1}]$

Let $b > 0$ be a fixed positive scalar. Show {$e^{2\pi i bnx}$} for $n \in \mathbb{Z}$ is a orthogonal (but not orthonormal) basis for $L^2[0,b^{-1}]$. I was able to show it is orthogonal and not ...
42 views

52 views

### Non compacity of a subset of $\ell^1$

Consider $l^1 = \left\{ \{a_n\}_{n=1}^\infty : a_n \in \mathbb{R}, \ \ \sum_{n=1}^\infty |a_n| < \infty \right\}$ and let $K = \left\{f \in l^1 : |f(k)| \leq \frac{1}{k} \ \ \forall k\right\}$. ...
27 views

### Generalisation: isomorphism of sequences spaces

As is well known, the isometric isomorphism $(c_0)^* \cong \ell^1$ holds. Is there an analogous statement for general $L^p$ spaces? Perhaps a good start would be to wonder about generalizations of ...
59 views

### $L^1$ convergent subsequence of increasing functions uniformly bounded in $L^2$

Problem: Suppose $f_n \colon [0,1] \to [0,\infty)$ for $n \in \mathbb{N}$ are increasing functions which are uniformly bounded in $L^2([0,1])$. Show that there exists a subsequence which ...
75 views

37 views

### Verification of convergence of random variables

Let $(X_n)_{n \in \mathbb{N}}$ a series of random variables with: $$P(X_n = 2^n) = \frac{1}{2^n} \hspace{15pt}\text{and}\hspace{15pt} P(X_n = 0) = 1-\frac{1}{2^n}$$ for all $n \in \mathbb{N}$. ...
58 views

48 views

42 views

64 views

### $L^1(\mathbb R)$ convergence of min and max functions.

Let $f_k , f \in L^1(\mathbb R)$ , with $f_k,f \geq 0$ a.e. in $(0,1)$. Suppose that $f_k\rightarrow f$ pointwise a.e. in $(0,1)$ and that: $\int_0^1 f_n dx \rightarrow \int_0^1 fdx$ Prove that: ...
42 views

### What's the meaning of denoting $\ell^p$ as $\ell^p(\mathbb{N})$?

What's the meaning of denoting $\ell^p$ as $\ell^p(\mathbb{N})$? I read that it's like $\ell^p$ "over $\mathbb{N}$". But $l^p$ is sequences indexed by $\mathbb{N}$. So it seems weird to treat the ...
53 views

### Continuity of a function between $l^p$ spaces

Let us consider the function defined as $$F: l^4 \rightarrow l^6 \\ (x_1, \dots,x_n, \dots) \mapsto (x_1^{20}, \dots, x_n^{20}, \dots)$$ I am asked to prove whether this function is continuous or ...
73 views

### Convergence in $L^{3/2}$ and in $L^2$

Let $(f_n)_n$ be a bounded sequence in $L^3(\mathbb R)$, such that $f_n\rightarrow f$ in $L^{3/2}(\mathbb R)$. Prove that $f_n\rightarrow f$ converges in $L^2\mathbb (R)$. I have and idea to first ...
82 views

### Critique my proof: Proving that $C[a,b]$ is not a Hilbert Space under $L^2$ norm.

So, I would appreciate any critiques I can get for my proof of the following problem. The Problem: Let $[a.b]$ be the closed, bounded interval of real numbers. Show that the $L^2[a,b]$ inner product ...
30 views

### Defining an “improper” Lebesgue integral?

I've always been bugged by examples of functions that are (improperly) Riemann integrable, but not Lebesgue integrable, like $\frac{\sin x}{x}$. While I understand perfectly well why such a function ...
51 views

### Find the orthogonal complement on $L^2[0,1]$ of all polynomials.

The problem: Determine the orthogonal complement on $L^2[0,1]$ to all polynomials. My approach and intuition thus far: I know for sure intuitively that the orthogonal complement would just be the ...
42 views

### Functional Space Inequality for Sobolev Space and Lp Space

Let $X = C_{0}(\Omega) := \{ u \in C(\overline{\Omega})\,|\,u|_{\partial\Omega}=0\}$ and define $F : X \to X$ as Lipschitz continuous function and $F(0) = 0$. Let $\Omega\subset \mathbb{R}^{N}$ be a ...
38 views

### Why is the conditional expectation of an $L^{p}$-function again in $L^{p}$?

Let $(\Omega, \mathcal{A},P)$ be a probability space and let $X,Y\colon \Omega \rightarrow \mathbb{R}$ be random variables. Furthermore, let $Y$ be $p$-integrable. Then why is the conditional ...
46 views

### Show that $(f_n)_n$ is relatively compact in $L^p$ space
Let $I=[0,1]$, $Q=I\times I$ and $(u_n),(v_n)$ bounded sequences in $L^2(I)$. Assume $x\mapsto u_n(x), x\mapsto v(x)$ are continuous and monotone non decreasing on $I$ for all $n\in\mathbb{N}$; define\...
In our current Measure Theory Class, we bought up the notion for a function $f:\mathbb R \to \mathbb R$ that is continuous to have a compact support, is equivalent to the fact that \$\overline{\{x \in \...