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To prove equality of two sets, show each set inclusion separately. If both inclusions hold, every element belongs to one set exactly when it belongs to the other; the axiom of extensionality gives equality. Each direction may use different hypotheses. For example, when $x\in A$, the condition $x\notin A\setminus B$ implies $x\in B$; without $x\in A$, that inference would be invalid.

Two disjoint compact subsets of a Hausdorff space have disjoint open neighbourhoods. For every $p\in P$, separation of a point and a compact set gives disjoint open sets $G_p\ni p$ and $H_p\supseteq Q$. Choose finitely many $G_p$ covering $P$, then take their union for $U$ and the matching finite intersection of the $H_p$ for $V$. The compactness of both subsets is used, once in the point-versus-compact set result and once in the outer finite selection. The ambient Hausdorff space need not be compact or a normal space.

Let $(u_n,v_n)\to (0,0)$ with both coordinates nonzero, $r_n=|u_n|/4$, $s_n=|v_n|/4$, and $\varphi (t)=\max (1-|t|,0)$. The displayed function equals one at every $(u_n,v_n)$ and zero on both coordinate axes. Each coordinate line off the corresponding axis meets only finitely many supports, so its restriction is a finite maximum of continuous functions. On the axes it is zero. Hence $B$ is separately continuous and disproves convergence of this sequence in any topology making every separately continuous function continuous. Isotropic radial bumps need not have this property when one coordinate approaches zero much faster than the other.

A finite-dimensional complex semisimple Lie algebra which is an ideal in a finite-dimensional Lie algebra has the displayed direct complement. The restriction of the Killing form of $M$ to $L$ equals the Killing form of $L$: the induced adjoint maps of elements of $L$ on $M/L$ vanish. This restriction is nondegenerate by the Cartan criterion for semisimplicity. Invariance makes its orthogonal complement an ideal. Nondegeneracy on $L$ gives a direct vector-space sum with that complement, and the bracket of the two ideals lies in their zero intersection, so they commute. The ambient algebra need not itself be semisimple.

Let $V$ be a finite-dimensional complex vector space and let $L$ be a Lie subalgebra of its traceless endomorphisms acting irreducibly. Its solvable radical $R$ is simultaneously triangularizable by Lie's theorem. The resulting finite set of diagonal characters is invariant under conjugation by $e^{tx}$ for $x\in L$, because this conjugation preserves the ideal $R$ and gives an equivalent representation. Each character varies continuously with $t$ in a finite set, hence is constant; differentiation gives $\lambda ([x,r])=0$.

A nonzero common $R$-eigenspace exists by Lie's theorem. The displayed vanishing makes it invariant under all of $L$: $r(xv)=\lambda (r)xv+\lambda ([r,x])v=\lambda (r)xv$. Irreducibility therefore makes it all of $V$. Thus $R$ acts by scalars, and the zero-trace condition forces those scalars to vanish. The matrix representation is faithful, so $R=0$. The traceless condition matters: adjoining scalar matrices to an irreducible semisimple algebra gives a reductive algebra with nonzero centre.

Let a semisimple complex Lie algebra $L={⨁}_i{L}_i$ act faithfully and irreducibly on a space $V$ of prime dimension. Complete reducibility for $L_1$ splits $V$ into isotypic components. The commuting other ideals preserve these components, so irreducibility leaves just one, of the form $W_1\otimes {\mathrm{Hom}}_{L_1}(W_1,V)$. Repeating with the other factors gives $V\cong {⨂}_i{W}_i$, with each $W_i$ an irreducible module for its simple ideal.

Faithfulness makes every factor nontrivial. A nonabelian simple Lie algebra has only the trivial one-dimensional module, so $\dim W_i\ge 2$. Their product is the prime $\dim V$, forcing just one simple ideal. Hence $L$ is simple. This is a representation-dimension obstruction, independent of the classification of root systems.

A point interaction concentrates a coefficient of a linear differential equation at a single interior point. On an interval $0<x<\ell$, with $0<c<\ell$, consider $L_{\alpha }y=y^{\prime \prime }+\alpha y(c){\delta }_c$. The domain consists of continuous functions that are twice differentiable away from $c$, with the required one-sided derivatives. Multiplication of the Dirac delta by $y$ means the unambiguous point value $y(c){\delta }_c$. It does not mean an undefined product of a delta with an arbitrary discontinuous function.

The equation is interpreted using distributional derivatives. Smooth forcing gives an ordinary equation on each side, with transmission conditions for a delta coefficient at the interaction. A delta potential in the one-dimensional Schrödinger equation is a physical example after choosing the corresponding sign and constants.

The third derivative is the derivative of the second derivative. On the interior of a cubic polynomial piece it is constant. A cubic spline can have different constants on adjacent pieces, so its third derivative need not be continuous across a spline knot.

The antiplane field $w=b\arg (x+iy)/(2\pi )$ has a smooth local gradient on the punctured plane and satisfies the differential strain compatibility condition there. Its circulation around the defect axis is nevertheless $b\ne 0$, excluding a globally single-valued displacement on that domain. A radial cut permits a branch with an axial jump $b$ between the two cut faces; joining faces with that mismatch is the Volterra construction of a screw dislocation with Burgers vector parallel to the axis. The stress has a singular core where linear elasticity requires microscopic or plastic regularization. This example separates a local differential integrability test from global closure around a hole.

Schistosoma haematobium is a blood fluke whose adults commonly inhabit venous pathways around the urinary bladder. Eggs crossing host tissues can cause damage and pass into urine. The damaging tissue stage and the adult worm’s location should be distinguished.^[ref]

On a bounded Lipschitz domain, the Poisson equation $-\Delta u=f$ with Robin boundary condition ${\partial }_{n}u+\beta u=0$ has the displayed weak formulation in $H^1(\Omega )$ when $\beta$ is a bounded boundary coefficient. Boundary integrals use Sobolev traces. It follows from Green's first identity, requiring only first weak derivatives of $u$. If $\beta$ is uniformly positive, testing the difference of two solutions against itself proves uniqueness: its gradient and its boundary Sobolev trace vanish. The Poincare inequality with a boundary trace also makes this a coercive bilinear form.

For $-(p{u}^{\prime }{)}^{\prime }=f$, $u(b)=0$ and $u^{\prime }(c)+\kappa u(c)=0$, integration by parts gives the displayed endpoint coefficient. Ordinary derivative data and conormal data $p(c)u^{\prime }(c)+\kappa u(c)=0$ are different boundary conditions. If $p$ is bounded above and below by positive constants and $\kappa \ge 0$, the Poincaré inequality on functions vanishing at $b$ proves continuity and coercivity. The symmetric form defines an energy norm, and Galerkin orthogonality gives exact best approximation in that norm.

For an orthonormal family in a Hilbert space, the squared norm of every finite partial sum is the sum of squared coefficients. Square summability therefore makes the net of finite partial sums Cauchy, proving unconditional convergence by completeness. If the orthonormal family is complete, Bessel's inequality makes the coefficients of each vector square summable, and the residual after summing is orthogonal to the whole family, hence zero. This recovers the orthonormal expansion and Parseval identity.

Assume $a+b$ and $a-b$ are nonzero on the contour. For the scalar jump $(a+b)w_{+}+(b-a)w_{-}=c$, divide by a nonvanishing canonical homogeneous solution $\tilde{w}$ to obtain an additive jump $\rho =c/((a+b){\tilde{w}}_{+})$. Its Cauchy integral gives a particular quotient, while every additional quotient $H$ with no jump is homogeneous freedom. The density $f=w_{+}-w_{-}$ therefore differs from the normalized particular density by $-2b{\tilde{w}}_{+}H/(a-b)$. Normalization, index and endpoint conditions determine the allowable $H$; dropping it without such conditions does not give the general solution.

A Cauchy integral solves a prescribed boundary difference, not a prescribed sum. If $W_{+}+W_{-}=g$ on $(0,c)$, choose $\rho (z)=\sqrt{z(z-c)}\sim z$ at infinity. Its boundary values satisfy ${\rho }_{-}=-{\rho }_{+}$, so $V=W/\rho$ has difference $V_{+}-V_{-}=g/{\rho }_{+}$. Thus $W=\rho [C(g/{\rho }_{+})+H]$, where $C$ is the Cauchy integral with coefficient $1/(2\pi i)$ and $H$ has no jump. Endpoint integrability and normalization at infinity determine permissible regularizations and homogeneous terms. Simply using $C(g)$ for a sum jump is incorrect.

For arc capacities $(c_{12},c_{13},c_{24},c_{34},c_{35},c_{46},c_{56})=(3,3,2,2,3,3,2)$, the feasible flow $(1,3,1,2,1,3,1)$ has value four. The displayed residual path has bottleneck one and traverses $4\to 3$ opposite to the original arc $3\to 4$. Augmentation cancels one unit there and gives $(2,3,2,1,2,3,2)$ of value five. The sink cut has capacity five, proving optimality. Reverse residual arcs are what permit rerouting instead of only adding flow along unused forward arcs.

A representation of a C-star algebra on a Hilbert space is a star-homomorphism $\pi :A\to \mathcal{B}(H)$. It is cyclic if some vector $\xi$ has dense orbit span as displayed; the orbit $\pi (A)\xi$ is itself a linear subspace. For a unital representation, a unit cyclic vector defines a state on a C-star algebra by $a\mapsto ⟨\pi (a)\xi ,\xi ⟩$. Conversely, the Gelfand–Naimark–Segal construction realizes every state this way. Cyclicity means density under the action of the whole algebra, not necessarily under powers of a single chosen element.

A T3 space is a regular topological space which is also a T1 space. Every metric space has this property: singletons are closed, and a sufficiently small ball about a point outside a closed set has closure avoiding that set. A second-countable T3 space is a metrizable space by Urysohn's metrization theorem.