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An integrable kernel is uniquely determined by its convolution operator on $L^2$. Convolve it with a smooth compactly supported approximate identity ${\eta }_{\epsilon }$, whose members also lie in $L^2$. A zero operator makes $h\ast {\eta }_{\epsilon }=0$, while translation continuity in Lp gives $h\ast {\eta }_{\epsilon }\to h$ in $L^1$, proving the assertion. Since the adjoint operator of convolution by $f$ is convolution by $f^{♯}(s)=\overline{f(-s)}$, this also proves the exact self-adjointness criterion $f=f^{♯}$ almost everywhere, including necessity at the level of kernels.

For finite complex measures on the additive circle group, convolution is the pushforward measure of their product under addition modulo $2\pi$. EquivalentlyBounded tests are integrable against the product variation measure, so Fubini's theorem applies. Testing the factorization $e^{-in(x+y)}=e^{-inx}{e}^{-iny}$ proves the displayed Fourier coefficient of a measure on the circle identity. The convolution of two probability measures is again a probability measure. On these measures convolution preserves simultaneous convergence in the weak-star topology: multiply the converging coefficients and use their common unit variation bound and dense trigonometric tests.

Tonelli's theorem gives the product integral of $|g(x-y)h(y)|$ as $\Vert g{\Vert }_1\Vert h{\Vert }_1$. Fubini's theorem then defines the convolution almost everywhere and gives an integrable result with norm at most that product. Multiplication by a complex exponential of modulus one preserves absolute integrability and permits the usual Fourier transform convolution identity.

Every set of at least $d+2$ points in ${\mathbb{R}}^d$ can be partitioned into two nonempty parts with intersecting convex hulls. To see this, take a nonzero affine dependence on $d+2$ of its points. Its positive and negative coefficient sums have the same magnitude $L>0$. Dividing by $L$ expresses one point as a convex combination of the positive-coefficient points and also as a convex combination of the negative-coefficient points. Distribute zero-coefficient and remaining points arbitrarily between the parts. This theorem obstructs arbitrary labels for affine linear classifiers.

Start from a nondegenerate tetrahedron and insert remaining points one at a time. Discard a point inside the current convex polyhedron; otherwise replace faces visible from it by the cone from that point to the horizon of a convex polyhedron. Scanning every current face gives $O(n^2)$ worst-case time complexity and $O(n)$ current-hull storage in three dimensions. Coplanar input requires a separate degeneracy policy.

For a differentiable real function $g$, a point of convexity is a point $v_0$ whose tangent line lies below the entire graph. The function need not be convex at other points. The Weierstrass excess function is exactly the difference between the function and this tangent line when applied to a variational integrand as a function of slope.

For a globally Lipschitz nonlinearity of constant $ϵ$, the map taking $v$ to the zero-Dirichlet Poisson solution with load $f(v)$ is a contraction in the gradient norm if the displayed inequality holds. Poincaré gives the factor $C_P$ from the Sobolev input to $L^2$, and testing the linear Poisson equation gives another factor $C_P$ back to the gradient norm. The condition yields a unique weak solution without requiring $f(0)=0$. With the optimal Poincaré constant it reads $ϵ<{\lambda }_1$.

Suppose all distances lie in $[0,D]$, $\psi$ is continuous, $0<\psi (r)<r$ for positive $r$, and ${\lim }_{r\to 0}\psi (r)/r<1$. The ratio is uniformly bounded below one near zero, and its maximum on the remaining compact scalar interval is also below one. Thus one constant $q<1$ works on the entire bounded set. If the set is complete and the map preserves it, Banach's theorem gives a unique fixed point. Compactness of the set itself is not required.

If the continuous dual space $X^{\prime }$ of a normed vector space is norm separable, then $X$ is separable. Choose a dense sequence $f_n$ in the dual unit ball and unit-ball points $x_n$ with $|f_n(x_n)|\ge \Vert f_n\Vert /2$. If their closed linear span were proper, the Hahn-Banach distance formula would give a norm-one functional $f$ vanishing on it. An $f_n$ within $1/4$ of $f$ would simultaneously satisfy $|f_n(x_n)|<1/4$ and $|f_n(x_n)|>3/8$, a contradiction. Rational linear combinations give a countable dense subset of $X$.

The converse fails by duality of l1 and l infinity: ${\ell }^1$ is separable, whereas ${\ell }^{\infty }$ contains uncountably many zero-one sequences at mutual distance one.

A context-free language recognized by scanning a string over the alphabet $\{0,1\}$ from right to left, counting 0 as $+1$ and 1 as $-1$, and requiring every partial count to remain nonnegative. A context-free grammar with a unique parse tree for each generated string is $S\to ϵ\mid S0\mid S1B0$, $B\to ϵ\mid B1B0$. To see uniqueness, reverse the strings: every nonempty nonnegative path either never returns to zero after its first step, or has a unique first return. These two cases determine the production and its component substrings.

A point of the Euclidean plane is constructible if it can be obtained from $(0,0)$ and $(1,0)$ by finitely many straightedge-and-compass operations: draw a line through two obtained distinct points, draw a circle with obtained centre and radius equal to an obtained segment length, and retain isolated intersection points. Its two coordinates are constructible numbers. Conversely, constructible real coordinates can be placed on perpendicular axes and their coordinate lines intersected, giving the point. Coordinates created at each isolated intersection require only solving linear or quadratic equations over the current coordinate field; hence they lie in a real field extension obtained by a finite tower of quadratic extensions. Coincident curves supply no isolated new intersection points.

The critical cone consists of linearized feasible directions along which the objective function has zero first-order change. For constraints $c_i\ge 0$, its directions satisfy $D{c}_{i}d\ge 0$ at active constraints and $\nabla f^{T}d=0$; equality constraints impose zero linearized residual. Under the usual regularity conditions, second-order Karush-Kuhn-Tucker conditions test the optimization Lagrangian Hessian matrix on this cone. Positive active multipliers force their corresponding directional residuals to vanish.

An L1 penalty can enforce a constraint exactly at a finite weight, rather than only in an infinite-weight limit. For a convex equality-constrained problem with an optimal multiplier $y_{\ast }$ and optimization Lagrangian $f-y^{T}c$, the lower bound $f(x)-y_{\ast }^{T}c(x)\ge f(x_{\ast })$ givesThus $\rho >\Vert y_{\ast }{\Vert }_{\infty }$ makes every penalty minimizer feasible and optimal. Local versions require appropriate regularity and second-order assumptions. The nonsquared norm is essential to this finite-threshold property; a quadratic penalty method usually approaches feasibility only as its weight diverges.

Sequential quadratic programming computes steps from a quadratic model of the optimization Lagrangian with linearized constraints. Its constrained Newton system couples a primal step and multiplier estimates. A line search or trust region then controls the use of that local model.

For $\dot{N}=N[P(N)-E]$ and $P(N)={\prod }_{i=1}^3(a_i-N)$ with $0<a_1<a_2<a_3$, extinction changes stability at $E_T$, while the upper stable population and an unstable separator merge at $E_M$. If $E_T<E_M$, an interval of efforts supports stable extinction and a stable positive population simultaneously; at $E_M$ the positive state collapses. If the thresholds have the opposite order, the high-population pair disappears before the lower stable branch reaches extinction. At a positive equilibrium stability follows from the sign of $N{P}^{\prime }(N)$.

Use the arcs and points of the noncompact counterexample to the Šura-Bura lemma, and add the central ray $\{(1/2,t):t\ge 1\}$ to form $X$. Define $C_n=\{p,q\}\cup {\bigcup }_{m\ge n}{A}_m\cup \{(1/2,t):t\ge n\}$. Each graph tail is connected because its arcs meet the ray. Its relative closure in $X$ adds just $p,q$, so each $C_n$ is closed and connected. These sets decrease and have intersection $\{p,q\}$, which is compact and disconnected. The ambient $X=C_1$ is connected and Hausdorff but noncompact.