113 problems.
- Inner approximation of a finite-measure set by compact subsets
- Complete totally bounded subsets of a metric space are compact
- Fourier transform is injective on \(L^1\), but inversion need not hold pointwise
- If \(f_n\in C^1[a,b]\) with \(f_n'\to g\) uniformly and \(f_n(x_0)\) convergent, then \(f_n\to f\) uniformly with \(f'=g\)
- Continuity of outer measure on Carathéodory-measurable sets
- \(\|f\|_p\to\|f\|_\infty\) as \(p\to\infty\) on finite measure spaces
- \(f=0\) a.e. iff \(\int_E f=0\) for every measurable \(E\)
- Countable unions of null sets are null
- Measurable sets are \(F_\sigma\) plus null and \(G_\delta\) minus null
- \(f,g\in L^1\) implies \(f\ast g\in L^1\) and \(\|f\ast g\|_1\leq\|f\|_1\|g\|_1\)
- Convolution of an \(L^1\) function with a bounded function is bounded and uniformly continuous
- Open subsets of \(\mathbb{R}\) are countable unions of disjoint open intervals
- Translation is continuous in \(L^1\)
- Every compact set in a metric space is closed and bounded
- \(f\in L^1\) need not imply \(\hat f\in L^1\); if both, then \(f\) is bounded, uniformly continuous, and vanishes at infinity
- Uniform limits, differentiability counterexamples, and the Cantor set
- \(\partial_i(f\ast g)=f\ast\partial_i g\) for \(f\in L^1\)
- Continuous functions on \([0,1]\) with vanishing moments are identically zero
- Cylinder functions of measurable functions are measurable
- Reverse triangle inequality, \(\sup\) and \(\inf\), and the Archimedean property
- Translation is continuous in \(L^p\) for uniformly continuous functions
- Almost everywhere and uniform convergence of approximations to the identity
- Dilation invariance of the Lebesgue integral
- Middle-thirds Cantor set is compact, totally disconnected, perfect, and null; Borel–Cantelli lemma
- Compact support and vanishing at infinity of convolutions
- Every open set in \(\RR^n\) is a countable union of almost disjoint closed cubes
- A pairwise disjoint family of intervals with nonempty interior in \(\RR\) is countable
- Uniqueness of \(L^1\) Fourier transforms and \(\int\hat f\,g=\int f\,\hat g\)
- Translation invariance of Lebesgue integral, \(L^{1}\) continuity, and regularity of measurable sets
- Convergence of integrals does not imply an integrable dominant
- \(\|f\ast\phi_t-f\|_1\to 0\) as \(t\to 0\) for an approximate identity \(\phi\)
- Translation invariance of the Lebesgue integral
- No function continuous on \(\QQ\) and discontinuous on \(\RR\setminus\QQ\)
- Disjoint compact and closed sets have positive distance
- \(L^p\) inclusions on finite measure spaces and \(\ell^p\) inclusions on \(\mathbb{Z}\)
- Fourier multiplication formula fails for unbounded \(g\); \(C^1\) functions equal their Fourier series
- \(\lim_{p\to\infty}\|f\|_p=\|f\|_\infty\) on a finite-measure space
- \(\lim_{n\to\infty}\sum_{k\ge 1}k^{-2}\sin^n(k)\) and \(\lim_{n\to\infty}\sum_{k\ge 1}k^{-1}e^{-k/n}\)
- Uniform limits preserve boundedness and continuity
- Uniform, pointwise, and a.e. convergence, and uniform convergence of \(\sum x^n/n!\) on compacta
- The sum of countably many measures is a measure
- The Fourier transform of an \(L^1\) function is bounded and uniformly continuous
- Integration by parts, special case
- Vanishing of \(\int f\), integrability of bounded functions, and density of simple, step, and \(C_c^\infty\) functions in \(L^1\)
- Convolution of continuous compactly supported functions is continuous and compactly supported
- Markov’s inequality and \(\int|f|=0\) if and only if \(f=0\) a.e.
- Support, uniform continuity, vanishing at infinity, and derivatives of convolutions
- Lebesgue regularity, invariance of the integral, convolution, and \(L^p\) norms
- Continuity, uniform convergence of series, Cesàro means, and limits of integrals
- Integrability of a radial power
- Translation invariance of Lebesgue measure and of the integral
- Hölder, completeness, a.e. uniform convergence, and density of simple functions in \(L^\infty\)
- Completeness, compactness, and Heine–Borel in metric spaces
- Convolution with an approximate identity converges in \(L^1\), almost everywhere, and uniformly
- Measurability of \(f(x)\) and \(f(x-y)g(y)\) on \(\RR^n\times\RR^n\), and \(\|f*g\|_p\le\|g\|_1\|f\|_p\) for \(p=1,2,\infty\) when \(f\in L^1\cap L^\infty\) and \(g\in L^1\)
- Continuity of \(d(x,F)\) and finiteness of \(\int\delta(y)/|x-y|^2\,dy\)
- Approximating a bounded set by open and closed sets, equivalent to measurability
- Limit interchange for derivatives and integrals; closed subsets of metric spaces are complete
- Countable disjoint intervals, open sets in \(\RR\) and \(\RR^n\), the Cantor set, and Borel–Cantelli
- Product rule from the definition, and Riemann integrability with finitely many discontinuities
- The maximal function of an \(L^1\) function need not be locally integrable
- Power series \(\sum a_n x^n\) and \(\sum b_n x^n\) with radii of convergence \(R_1\) and \(R_2\)
- \(L^1\) convergence of an \(L^2\) sequence: the limit, a.e. failure, and an a.e. subsequence
- Convolution of \(L^1\) functions, vanishing at infinity, and the Fourier inversion formula
- Continuous functions on compact sets are uniformly continuous
- Uniform continuity, sets of discontinuities, and non-uniform limits of continuous functions
- Radius of convergence of \(\sum (a_n+b_n)x^n\)
- von Neumann’s mean ergodic theorem for a unitary operator
- Bessel’s inequality and reconstruction from an orthonormal sequence
- The Weierstrass approximation theorem
- Uniform limits of bounded and continuous functions, termwise differentiation, and \(\sum x^n/n!\) on compacta
- Tonelli’s theorem and the layer-cake representation of a nonnegative measurable function
- Radius of \(\sum a_n b_n x^n\) at least the product of the radii of \(\sum a_n x^n\) and \(\sum b_n x^n\), strictly in an example
- \(L^1\) functions are finite a.e., absolutely summable series in \(L^1\), and a dominated-convergence limit
- The Kronecker sequences \(u_k(j)=\delta_{kj}\) form an orthonormal system in \(\ell^2(\ZZ)\)
- No function continuous exactly on \(\QQ\)
- Uniform convergence \(f*\phi_t\to f\) as \(t\to 0\) for bounded uniformly continuous \(f\) when \(\int\phi=1\)
- Fatou’s lemma and summing series in \(L^1\)
- \(L^1((0,2\pi))\) functions as \(g+h\) with \(g\in L^2\) and \(\|h\|_1\) arbitrarily small
- \(\int(f+g)=\int f+\int g\) on \(L^+\), countable additivity, and \(\mu_f(E_j)\to\mu_f(E)\) when \(E_j\nearrow E\)
- Sequences, uniform convergence, and Bolzano-Weierstrass
- Riesz representation for \(L^2\)
- Translation invariance of Lebesgue measure and integrals
- \(\|a\|_{\ell^q}\le\|a\|_{\ell^p}\) for \(0<p<q\le\infty\)
- Young’s convolution inequality
- \(L^\infty(\RR^n)\) is a Banach space, and \(L^1\cap L^\infty\subset L^2\) with \(\|f\|_2\le\|f\|_1^{1/2}\|f\|_\infty^{1/2}\)
- Linear functionals: definition, boundedness equivalent to continuity, and completeness of the dual
- \(\sup_{y>0}|f*P_y|\le C\,Hf\) and \(f*P_y\to f\) a.e. for the Poisson kernel
- Best linear approximation to \(x^2\) in \(L^2([0,1])\) by projection onto \(1\) and \(\sqrt3(2x-1)\)
- Compact sets are closed and bounded, complete totally bounded sets are compact, and \(\dist(K,F)>0\)
- \(\nu\perp\mu\) and \(\nu\ll|\mu|\) implies \(\nu=0\)
- \(\widehat G=\bigl(\frac{\sin\pi\xi}{\pi\xi}\bigr)^2\) for the tent function \(G\); \(\widehat F\); an \(L^1\) Fourier transform not in \(L^1\)
- Measurability of \(\inf_k f_k\) and \(\sup_k f_k\), Fatou’s lemma, and the monotone convergence theorem from Fatou
- Continuity of outer measure, vanishing integrals, and density in \(L^1\)
- \(L^q\subseteq L^p\) for \(p<q\) implies no sets of arbitrarily large finite measure
- \(\int_0^\infty\frac{g(x)}{x}\int_0^x f(y)\,dy\,dx\le AB\) with \(A=\int_0^\infty f(y)y^{-1/2}\,dy\) and \(B=\|g\|_2\)
- Hölder’s inequality, Minkowski’s inequality, \(L^\infty[0,1]\), and \(\lim_{p\to\infty}\int_{[0,1]}f^p\)
- Countable subadditivity and outer regularity of Lebesgue outer measure
- The first Borel–Cantelli lemma
- \(L^p\) norms tend to \(L^\infty\), the converse of dominated convergence, translation continuity, and \(L^p\) inclusions
- \(Hf(x)\ge\frac{c}{(1+|x|)^n}\) for \(0\neq f\in L^1(\RR^n)\)
- \(E^\perp\) is a closed subspace of a Hilbert space and \((E^\perp)^\perp=\overline{E}\)
- \(E\) is measurable if and only if \(E\) is regular
- Layer-cake formula for a measurable function
- Fourier transforms of \(L^1\) functions: boundedness, uniqueness, the multiplication formula, and Fourier series of \(C^1\) functions
- The supremum and pointwise limit of measurable functions are measurable
- Functions in \(L^p(0,\infty)\) precisely for \(a<p<b\), for \(a\le p\le b\), and for \(p=a\)
- \((1+|\xi|^2)^{-\epsilon}\) is the Fourier transform of an \(L^1\) function
- – What does it mean for a series to converge? How can you check this?
- If \(\lim_{x\to\infty}f(x)\) and \(\lim_{x\to\infty}f'(x)\) exist, then \(\lim f'=0\)
- Bessel’s inequality and reconstruction from an orthonormal sequence
- \(\|g\|_p\to\|g\|_\infty\) as \(p\to\infty\) on \([0,1]\), and \(\|\Lambda_g\|_{(L^1)^*}=\|g\|_\infty\)
- Hölder’s inequality \(\bigl\|\prod f_j\bigr\|_r\le\prod\|f_j\|_{p_j}\) when \(\sum 1/p_j=1/r\le 1\)