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Every continuous real-valued function on a closed subspace of a normal space extends continuously to the whole space. If its values lie in a closed interval, the extension can be kept in that interval.

If three nonproportional linear forms exclude three distinct residue classes modulo every sufficiently large prime, the upper-bound sieve has dimension three and gives an upper bound of order $x/(\log x{)}^3$.

For an upper Hessenberg matrix iterate, when an off-block subdiagonal entry tends to zero, QR iteration asymptotically separates the corresponding invariant spectral blocks even if eigenvalues within one block have equal modulus.

The accumulated orthogonal factor in QR iteration is the $Q$ factor of a matrix power, so its leading columns span the same spaces as simultaneous power iteration.

An unshifted QR step preserves the bandwidth of a real symmetric banded matrix. A Givens-rotation implementation exposes this as bulges created during triangularization and removed when the factors are multiplied in reverse order.

LetThenThus ${\overline{Q}}_k{\overline{R}}_k$ is a QR decomposition of the matrix power $A^{k+1}$. In particular, the first $r$ columns of ${\overline{Q}}_k$ span the same subspace as the first $r$ columns of $A^{k+1}$ whenever those columns are independent.

For a finite Galois extension $L/K$ of non-Archimedean local fields, normalize $v_L$ to have value group $\mathbb{Z}$. For integers $i\ge 0$, the lower ramification groups areThe group $G_0$ is the inertia group, whose automorphisms fix the residue field. The group $G_1$ is the wild inertia group, a group of prime-power order for the residue characteristic. Larger indices demand closer agreement with the identity on integral elements.

An unramified extension has $G_0=1$. A nontrivial totally and tamely ramified extension has $G_0=\mathrm{Gal}(L/K)$ and $G_1=1$. In a wild extension further groups can survive, and their sizes compute the different exponent from ramification groups. The upper ramification numbering reindexes this filtration to make it compatible with quotient Galois groups.

Let $X$ have contractible universal cover, let $K$ be a path-connected simplicial complex, and let $i:K^1\hookrightarrow K$. A map $f:|K^1|\to X$ extends to $|K|$ exactly when its induced fundamental-group map factors through $i_{\ast }$. The factorization kills every 2-simplex boundary; all higher-dimensional boundary maps lift to the contractible universal cover and are null-homotopic.

For a path-connected locally path-connected base, any two based universal covers lift to one another. The two composites and the respective identity maps are based lifts of the same covering maps, so uniqueness of lifts makes the composites identities. The lift is therefore a homeomorphism.

Rational homology satisfiesIt records the free rank of integral homology and discards torsion.

The determinant $U(n)\to S^1$ sends a unitary loop to a circle-valued loop. A nonzero determinant winding number proves that the original loop class has infinite order.

The Werner–Holevo channel is $T(\rho )=(\mathrm{Tr}(\rho )I-{\rho }^T)/(d-1)$. Its normalized Choi matrix is the maximally mixed state on the antisymmetric subspace.

The normal derivative of a differentiable function $u$ along a hypersurface with unit normal $N$ is the directional derivativeHigher normal derivatives at a point $x$ may be defined by differentiating the one-variable function $s\mapsto u(x+sN(x))$ repeatedly at $s=0$.

The union-closed sets conjecture states that every finite nontrivial union-closed family has an element contained in at least half of its members.

An operator $T$ is a unilateral shift exactly when it is an isometry, $(\mathrm{Im}T{)}^{\perp }$ is one-dimensional, and ${\bigcap }_{n\ge 1}\mathrm{Im}(T^n)=\{0\}$. A unit vector $e_1$ in the first orthogonal complement is wandering, and $e_n=T^{n-1}{e}_1$ is a complete orthonormal basis.

For a bounded linear operator $T:X\to X$ on a complex Banach space, the point spectrum is precisely its set of eigenvalues:Here $I$ is the identity operator. Thus membership requires an actual nonzero eigenvector, whereas the spectrum of a bounded operator only requires failure of a bounded inverse. Every eigenvalue belongs to the spectrum, but the inclusion can be strict. The multiplication operator discussed below has an entire interval in its spectrum and empty point spectrum.^[ref]

If two tests have the same size and each has strictly greater power than the other at some alternative, neither test is uniformly most powerful.

For $u\in C^2(\Omega )\cap C^0(\overline{\Omega })$ and a uniformly elliptic second-order operator with bounded coefficients $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$ with $c\le 0$, the inequality $Lu\ge 0$ on a bounded domain impliesAn exponential barrier reduces the non-strict inequality to a contradiction at a positive interior maximum.

A second-order operator is strictly elliptic at a point when its symmetric principal coefficient matrix is positive definite there. Uniform ellipticity strengthens this pointwise condition by requiring one positive lower bound throughout the domain.

If ${\mathbb{C}}^{\ast }$ embeds holomorphically in a compact Riemann surface $R$, the embedding extends across zero and infinity to a degree-one holomorphic map from the Riemann sphere to $R$. Hence $R$ is conformally the sphere.