1
vote
1answer
27 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
0
votes
1answer
41 views

Inequality important in $L^p$ space

If$\,\,$ $0<p<\infty$, put$\,\,$ $\gamma_{p}=\max(1,2^{p-1})$, and show that $$|\alpha-\beta|^p \leq \gamma_{p}(|\alpha|^p + |\beta|^p)$$ for arbitrary complex numbers $\alpha$ and $\beta$. ...
4
votes
3answers
56 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
0
votes
1answer
27 views

On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
0
votes
1answer
34 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
0
votes
1answer
66 views

Hilbert space $L^{2}(0,\pi)$

I wanted to know how I should proceed if I wanted to prove that the closed subspace of $L^{2}(0,\pi)$ generated by {$\sin(kx): k=1,2,...$} coincides with $L^{2}(0,\pi)$. Thanks.
10
votes
2answers
105 views

Finite dimensional subspace of $C([0,1])$

Let $S$ be a subspace of $C([0,1])$, i.e. the continuous real functions on $[0,1]$. Assume that there exists $c>0$ such that $\|f\|_\infty\leq c \|f\|_2$ for all $f\in S$. Then $S$ must be ...
3
votes
0answers
26 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
2
votes
1answer
33 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
0
votes
1answer
25 views

Interpolation inequality in $L^p$ space

Let $f \in L^{p}(E) \cap L^{q}(E)$ with $p<q$. How to prove that $f \in L^{h}(E)$ for every $h \in (p,q)$ and the following interpolation inequality: $||f||_{h} \leq ||f||_{p}^{\frac{p}{h}} + ...
1
vote
1answer
42 views

Functional Analysis, $L^p$ spaces

Can anyone help me finding for which $p$ the function $$f(x)=\frac{1}{x^{\alpha}+x^{\beta}} \in L^{p}(0,\infty)$$ where $0<\alpha \le \beta<\infty$ are given. Thanks a lot.
1
vote
0answers
11 views

$f \mapsto \int f^2$ is $L^1$-weakly lower semicontinuous

If $f \in C(\mathbb [0,1], \mathbb R)$ is $$ f \mapsto \int_0^1 f(t)^2\ dt$$ $L^1$-weakly lower semicontinuous? I.e. if $$\int_0^1 f_n g \rightarrow \int_0^1 f g$$ for every $g \in L^{\infty}$, then ...
2
votes
1answer
93 views

$\ell^{\infty}(\mathbb N)$ is not a separable space

I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable. My attempt Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
0
votes
1answer
33 views

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm?

Let $f \in L^1[0,1]$, can we approximate $f$ by a polynomial, in sup norm ? I know that the algebra of polynomials is dense in algebra of continuous functions, wrt to sup norm, And I know that if $f ...
0
votes
0answers
31 views

Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
0
votes
2answers
48 views

Given that $f \in L^{p_0} \cap L^{\infty}$ show $f \in L^p$ for all $p_0 \leq p \leq \infty$

Title says it all. I feel like Holder's inequality may be useful here but I'm struggling on where to start. Not looking for a solution, just some tips to jump start from.
3
votes
1answer
72 views

Are $L_p$ spaces of functions with separable support separable?

Let $X$ be a separable space. Is $L_p$$(X, \mu, V)$ a separable space? Here, $(V, |\cdot|_V)$ is a normed space. And a norm of $L_p(X, \mu, V)$ is: $$ \|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p ...
0
votes
1answer
65 views

Cauchy–Schwarz inequality on vector-valued L2 space

Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking ...
2
votes
1answer
64 views

If $u_n \rightharpoonup u$ in $L^2(\Omega)$ and $u_n^2 \rightharpoonup v$ in $L^1(\Omega)$ then is $v=u^2$?

If $u_n \rightharpoonup u$ in $L^2(\Omega)$ and $u_n^2 \rightharpoonup v$ in $L^1(\Omega)$ then is $v=u^2$? We assume that the domain $\Omega$ is bounded. If not is there any way to ensure this?
1
vote
1answer
34 views

Equivalent condition for equi-integrability

I am looking for a Lemma that gives an equivalent formulation for a family of functions to be equi-integrable: is it true that if $\{f_j\}_j\in L^1$, then we can write $f_j=f^1_j+f_j^2\in L^1+L^p$, ...
-1
votes
1answer
36 views

Prove that $f \ast g$ is continuous and bounded if $f\in L^1(R^n)$ and $g\in L^\propto (R^n)$ [duplicate]

My Engliah is no so good and it is my first time to use this website, so I apologize for it if I didnot make myself clearly:)
2
votes
2answers
28 views

Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
3
votes
2answers
87 views

Why is the set of all $\infty$-tuples with finitely many non-zero rational terms dense in $\ell_2$?

This statement has been given as an example in the book "Introductory real analysis" written by Kolmogorov and Fomin: The set of all points $x=(x_1,x_2,\cdots,x_n,\cdots)$ with only finitely ...
1
vote
1answer
39 views

Approximation of bounded Borel functions

Let $K$ be a compact space and let $B$ be the space of bounded Borel functions on $K$ equipped with the supremum norm. Show that simple functions (i.e. functions attaining only a finite number of ...
1
vote
1answer
56 views

Inclusion of $l^p$ space for sequences

Inclusion of $L^p$ spaces for functions has been discussed here. Does this apply to $l^p$ space of sequences similarly? I tried to show the following: For $1\leq p<q<\infty$, $l^q\subset l^p$ ...
1
vote
1answer
126 views

Limit of $\|x\|_p$ as $p\rightarrow\infty$ [duplicate]

I am not sure how to start this hw problem. Here it is: Let $x$ be a given vector in $\ell^p$ with $1\le p\lt\infty$. Show that $x \in \ell^\infty$ and $$\lim_{p \to \infty} ||x||_p = ...
3
votes
1answer
39 views

$L^{p}$ inequality with a lower bound on measure

I am working on the following problem: Suppose $f \in L^{p}(X)$ for some $0 < p < \infty$ and the space $X$ is such that each set of positive measure has measure $\geq m$ for some $m > 0$. ...
0
votes
1answer
40 views

Show that H$(I)$ is a closed subspace of $L^2(I)$

EDIT: Original statement is not true, added condition. Let $I$ be the unit interval, define $H(I) = \{f\in AC(I)$ and $f'\in L^2(I)\}$. I want to show that $H(I)$ a closed subspace of $L^2(I)$. ...
2
votes
2answers
56 views

$\int_{\mathbb R} |f(x)| dx < \infty \implies \sum_{n\in \mathbb Z} |f(n)| < \infty $?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
3
votes
1answer
57 views

Compact kernel operator on $L^p$ space

Let $\displaystyle U_1 \subset \mathbb R^{n_1}$ and $\displaystyle U_2 \subset \mathbb R^{n_2} $ measurable sets, $\displaystyle 1 < p,q < \infty $ and consider the measurable function ...
1
vote
0answers
68 views

Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ a.e. on D.

I want to prove this result: Let $D$ be an open subset of $\mathbb{R}^n$, $p \in[1,\infty)$ and $f$ be in $L^p(D)$. Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ ...
-1
votes
1answer
29 views

The convergence in $L^{p_1}$ and $L^{p_2}$

Suppose $f_k$ is a sequence of $M-$measurable function. Let $p_1$ and $p_2\in[1,\infty)$,and $f_k\in L^{p_1}\cap L^{p_2}$.Also suppose $\exists g\in L^{p_1}$ and $h\in L^{p_2}$ s.t. $f_k\rightarrow g$ ...
0
votes
1answer
38 views

Prove that if $f,g \in L^1(-\infty,+\infty)$ and $h(x)=\int^{+\infty}_{-\infty}f(x-e^y)g(y)dy$ then $h \in L^1(-\infty,+\infty)$

Prove that if $f,g \in L^1(-\infty,+\infty)$ and $$h(x)=\int^{+\infty}_{-\infty}f(x-e^y)g(y)dy$$ then $h \in L^1(-\infty,+\infty)$ and $$||h||_1 \leq ||f||_1 ||g||_1$$ I would be thankful if I see a ...
1
vote
3answers
46 views

Suppose $E \subset \mathbb{R}$ has infinite measure, and $f\in L^1(E)$, Is this true that $f \in L^{\infty}(E)$ necessarily?

Suppose $E \subset \mathbb{R}$ has infinite measure, and $f\in L^1(E)$, Is this true that $f \in L^{\infty}(E)$ necessarily ? I could not find a counterexample so far and it is a useful fact if it ...
0
votes
1answer
130 views

Continuous functions vanishing at infinity is always integrable?

Let $$C_{0}(\mathbb R)= \big\{\,f:\mathbb R \to \mathbb C\,\,\, \text{continuous and}\,\, \lim_{x\to \pm \infty}f(x)=0 \big\}.$$ Assume that $f\in C_{0}(\mathbb R)$. My question is: Is it always ...
2
votes
1answer
49 views

Subspace of Tempered Distributions

Let ${S_{h}}'(\mathbb{R}^{n})$ be the space of tempered distributions such that if $u\in {S_{h}}'(\mathbb{R}^{n})$, then $\lim_{\lambda\rightarrow \infty}{||\phi(\lambda D)u||_{\infty}} = 0$ for all ...
0
votes
1answer
44 views

$u$ in $L^p$ implies that $u+c$ is in $L^p$.

Suppose that $\displaystyle\int_U|u(x)|^p\,dx<\infty$, where $U\subset\mathbb{R}^{n}$ is bounded and $p$ is a positive integer. How can I prove that $\displaystyle\int_U|u(x)+c|^p\,dx<\infty$? ...
1
vote
1answer
89 views

$L^p$ Dominated Convergence Theorem

I want to prove the $L^p$ Dominated Convergence Theorem which says : Let $\{ f_n \}$ be a sequence of measurable functions that converges pointwise a.e. on E to $f$. For$ 1 \leq p < \infty$, ...
2
votes
0answers
43 views

Is this possible to prove that $L^p[E]$ is complete without using the concept of “rapidly Cauchy” sequences?

Is this possible to prove that $L^p[E]$ is complete without using the concept of "rapidly Cauchy" sequences ? In Royden's book, this how it is done. I was curious if there is an alternative approach ...
0
votes
1answer
36 views

Upper bound for the norm of inverse Fourier tansform

Recall Hausdorff-Young inequality: For any $f\in L^p(\mathbb{R}^n)$, we have $||\hat{f}||_q\le ||f||_p$, where $p$ and $q$ are conjugate exponents and $p\in[1,2]$. It seems to me that it follows ...
7
votes
4answers
337 views

$\ell^p$ is not isometric to $\ell^q$

The problem is this: if $1\le p<q<\infty$ then $\ell^p$ and $\ell^q$ are not isometric (as Banach spaces). This is an exercise but I'd like to see an elegant proof.
3
votes
1answer
32 views

Prove that $f\in L^{1}\left(\Omega\right)$ and $\lim_{n\rightarrow\infty}\int(\left|f_{n}\right|-\left|f_{n}-f\right|)=\int\left|f\right| $.

Let $(f_n)$ be a sequence in $L^1(\Omega)$ such that: (i) $f_n(x) \rightarrow f(x)$ a.e. (ii) $(f_n)$ is bounded in $L^1(\Omega)$ i.e. $\left\Vert f_{n}\right\Vert _{L^{1}}\leq M\forall n $ Prove ...
2
votes
1answer
20 views

Is $L^{4}\left(X,\nu\right)$ contained in $L^{4}\left(X,\mu\right)$?

Let $\mu$ and $\nu$ be positive measures in $\left(X,\mathrm{\mathcal{M}}\right)$ . Assume that $\mu\left(E\right)\leq\nu\left(E\right)\forall E\in\mathcal{M}$ . Is ...
5
votes
1answer
102 views

Subsequence convergence in $L^p$

I recall a fact that for functions $f_1,f_2,\ldots\in L^1$ such that $\|f_n-f\|_1\rightarrow 0$ as $n\rightarrow\infty$, there exists a subsequence $f_{n_i}$ that converges to $f$ almost everywhere. ...
0
votes
2answers
73 views

Prove $\|f\|_{L^p}$ is not equivalent to $\|f\|_{\infty}$ in $C[a,b]$

Prove that in $C[a,b]$ the uniform norm is not equivalent to the $L^p$ norm for $(1\leq p < \infty)$ I am stuck on showing that the function below satifies the claim. I know that f is continuous ...
3
votes
2answers
66 views

Deducing that $f$ is in $L^p$

Suppose $f : \mathbb{R} \to \mathbb{R}$ is integrable and there is positive $K$ and $0<c<1$ such that $$\int_{B} \left\vert \:f(x)\right\vert \:\mathrm{d}x \leq Km(B)^{c}$$ for every Borel ...
3
votes
1answer
77 views

$L^2$ norm of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$f_r(\theta)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
2
votes
1answer
39 views

Integral of $L^2$ function is continuous

For $f\in L^2(\mathbb{R})$, denote $$s_N(x)=\dfrac{1}{2\pi}\int_{-N}^N\hat{f}(t)e^{ixt}dt.$$ I'd like to prove that the integral converges, and that $s_N$ is continuous. Since $f\in ...
0
votes
1answer
24 views

Identity approximation for functions in $C_0(\mathbb{R})$

Let $K\in L^1(\mathbb{R})$ satisfy $\int_\mathbb{R} K(x)dx=1$, and denote $K_a(x)=\frac1aK(\frac{x}{a})$ for $a>0$. We get trivially that $\int_\mathbb{R}K_a(x)=1$. Let $x\in\mathbb{R}$ and $f\in ...
7
votes
2answers
491 views

$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?