Intereting Posts

When a change of variable results in equal limits of integration
Correlation matrix from Covariance matrix
Finitely Presented is Preserved by Extension
Strong and weak convergence in $\ell^1$
Proving $rank(\wp(x)) = rank(x)^+$
Exterior Derivative vs. Covariant Derivative vs. Lie Derivative
One-variable continuity of one partial derivative implies differentiability?
What is the probability of rolling $n$ dice until each side appears at least once?
Effect of differentiation on function growth rate
Proving $x^4+y^4=z^2$ has no integer solutions
Showing the exponential and logarithmic functions are unique in satisfying their properties
Why does $\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}={\dfrac{\pi-1}{2}}$?
Invariance of strategy-proof social choice function when peaks are made close from solution
Solve $ x^2+4=y^d$ in integers with $d\ge 3$
How to prove a limit exists using the $\epsilon$-$\delta$ definition of a limit

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each $f\in X$, we need to show that there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

please tell me how to solve this one, I have no clue.

- Prove that $X^\ast$ separable implies $X$ separable
- A compact operator is completely continuous.
- Riemann integrals of abstract functions into Banach spaces
- $\ell_\infty$ is a Grothendieck space
- Fréchet differentiability from Gâteaux differentiability
- The space of Riemannian metrics on a given manifold.

- A positive “Fourier transform” is integrable
- Derive Fourier transform of sinc function
- How to prove this inequality in Banach space?
- Derivative of Fourier transform: $F'=F$
- Categorical Banach space theory
- About the order of the $L^1$ norm of the Dirichlet kernel.
- Banach Spaces: Uniform Integral vs. Riemann Integral
- Fourier Series Representation $e^{ax}$
- a generalization of normal distribution to the complex case: complex integral over the real line
- Question about computing a Fourier transform of an integral transform related to fractional Brownian motion

Let $T\colon X\to \ell^1$ defined by $T(f)=\{\widehat f(n)\}_{n=-\infty}^{+\infty}$. As $X$ and $\ell^1$ are complete and $T$ is linear, we just need to check that the graph of $T$ is closed.

Let $\{f_k\}\subset X$ such that $f_k\to 0$ in $L^1$ and $T(f_k)\to y$ in $\ell^1$. We have for all $n$ that $\widehat{f_k}(n)\to 0$ and $y_n=\lim_{k\to +\infty}\widehat{f_k}(n)$ (as convergence in $\ell^1$ implies coordinatewise convergence. We conclude that $y=0$.

- A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$
- What is the rank of the cofactor matrix of a given matrix?
- Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
- Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem
- Do all square matrices have eigenvectors?
- Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous
- How to prove $\lim_{n \to \infty} (1+1/n)^n = e$?
- Neat way to find the kernel of a ring homomorphism
- Applications of Weierstrass Theorem & Stone Weierstrass Theorem
- How to prove that $e = \lim_{n \to \infty} (\sqrt{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt{n\#} $?
- Eigenvalues of linear operator $F(A) = AB + BA$
- Advice on self study of category theory
- Existence and value of $\lim_{n\to\infty} (\ln\frac{x}{n}+\sum_{k=1}^n \frac{1}{k+x})$ for $x>0$
- Is there a simple proof for ${\small 2}\frac{n}{3}$ is not an integer when $\frac{n}{3}$ is not an integer?
- $p$-adic completion of integers