A basis vector is one of the vectors in a specified basis of a vector space. Its coordinate column in that basis has one entry equal to one and all other entries zero. Columns of a matrix of a linear map are the images of domain basis vectors, expressed in the chosen codomain basis.

An ordered basis is a basis equipped with an ordering of its basis vectors. The order fixes which position of a coordinate vector corresponds to each basis vector.

A normalized basic sequence generates a spreading model when the norms of all finite scalar combinations converge as the first index tends to infinity, uniformly over increasing choices of the indices. The limiting norm on finitely supported scalar sequences is unchanged by increasing relabellings of coordinates. Completing this norm gives a Banach space with its canonical basic sequence. A diagonal application of the finite-colour infinite Ramsey theorem produces such a model from a subsequence of every normalized basic sequence.

An unconditional basic sequence is a Schauder basis for its closed span whose expansions converge after arbitrary permutations or coordinate suppressions. Equivalently, the finite coordinate projections are uniformly bounded. Failure of this property gives pairs of disjointly supported unit vectors at arbitrarily small distance, obtained by separating a finite combination with a large coordinate projection.

For $K\ge 1$, let $T_K(X)$ consist of finite normalized sequences satisfying $K^{-1}\sum |a_i|\le \Vert \sum a_i{x}_i\Vert \le \sum |a_i|$ for every scalar combination. Its tree order measures the depth of finite copies of the unit vector basis of ${\ell }^1$. A separable Banach space containing no copy of ${\ell }^1$ has a countable index because every such tree is closed and well-founded. Countable well-founded trees can be used to construct separable reflexive spaces with arbitrarily high index, precluding a separable reflexive universal space.

Every bounded sequence in a Banach space has either a weakly Cauchy sequence as a subsequence or a subsequence equivalent to the unit vector basis of ${\ell }^1$. The real-scalar proof represents vectors as continuous functions on a compact dual ball and uses infinite Ramsey theory to select independent lower and upper level sets when pointwise convergence cannot be obtained.

For a finite-dimensional associative algebra over an algebraically closed field, the Ext quiver has a vertex for each simple module isomorphism class and $a_{ij}$ arrows $i\to j$. For a basic algebra with $P_i=A{e}_i$, applying ${\mathrm{Hom}}_A(-,S_j)$ to the projective-cover sequence gives $a_{ij}=\dim e_j(J/J^2)e_i$. Our path algebra convention multiplies consecutive arrows $\alpha :i\to j$, $\beta :j\to l$ as $\beta \alpha$, making left modules have arrow maps $V_i\to V_j$. The orientation convention must be stated when comparing other definitions.

For $B=\overline{\mathbb{C}[1,x]}$ inside $A$, the displayed set is the full spectrum of $x$ in $A$. On a bounded complementary component, the maximum modulus principle gives $|p(\lambda )|\le {\max }_{{\sigma }_A(x)}|p|\le \Vert p(x)\Vert$. Evaluation $p(x)\mapsto p(\lambda )$ therefore extends to a character of an algebra on $B$, showing $\lambda \in {\sigma }_B(x)$. Conversely the unbounded resolvent component survives by resolvent components in a closed unital subalgebra. In particular, if zero lies in that component, $x^{-1}$ belongs to the norm closure of polynomials in $x$.

For a compact set $K$ in the domain, a norming bounded linear functional at $f(z)$ reduces the assertion to the scalar maximum modulus principle. If $z$ lies inside $K$, use its connected component of the interior; that component's boundary lies in $\partial K$. If $z\in \partial K$, the inequality is immediate. No smoothness or connectedness assumption on $K$ is required.

For a Banach algebra-valued holomorphic function, set $p_n(w)=\Vert f(w{)}^{2^n}{\Vert }^{1/2^n}$. These continuous functions decrease to the spectral radius by the spectral radius formula. If $M={\sup }_{\partial K}r(f(w))$, the continuous functions $\max \{p_n,M\}$ on $\partial K$ decrease to the continuous constant $M$. The Dini theorem gives uniform convergence. Apply the norm maximum principle for a Banach-space-valued holomorphic function to each power, and then let $n\to \infty$. This proof does not assume continuity of spectral radius.

If $Y$ is a proper closed subspace of a normed space $X$ and $0<\alpha <1$, there is a unit vector $x\in X$ whose distance from $Y$ exceeds $\alpha$.

An infinite-dimensional Banach space is hereditarily indecomposable if no closed infinite-dimensional subspace is a topological direct sum of two infinite-dimensional closed subspaces. Equivalently, for any two infinite-dimensional closed subspaces $U,V$ and any $\epsilon >0$, some unit vectors $u\in U,v\in V$ satisfy $\Vert u-v\Vert <\epsilon$. A positive lower bound on such distances would make the addition map from $U\oplus V$ an isomorphism onto a closed subspace.

A closed vector subspace $Y$ of a Banach space $X$ is complemented if a bounded linear operator $P:X\to X$ satisfies $P^2=P$ and $P(X)=Y$. Equivalently, $X$ is the topological direct sum of $Y$ and a closed subspace: a projection gives $X=Y\oplus \ker P$, and the coordinate projection of such a decomposition is bounded.

A Banach space $X$ has cotype $q$, $2\le q<\infty$, if a constant $C$ independent of the number of vectors satisfies $({\sum }_j\Vert x_j{\Vert }^q{)}^{1/q}\le C(\mathbb{E}\Vert {\sum }_j{\epsilon }_j{x}_j{\Vert }^2{)}^{1/2}$ for independent symmetric Rademacher random variables. It prevents many large vectors from having a small random sum.

A Banach space has the Banach-Saks property if every bounded sequence contains a subsequence whose arithmetic averages converge in norm. The weak Banach-Saks property requires this only for weakly convergent sequences. The weak Banach–Saks theorem in a Hilbert space establishes that weak version in Hilbert spaces.

For a closed linear subspace $Y$ of a Banach space $X$, finite codimension means the vector quotient $X/Y$ has finite dimension. This definition does not subtract two infinite dimensions. A local stable manifold has the codimension of its tangent subspace. If the linearized phase space has a direct sum decomposition $X=E^s\oplus E^u$ with $\dim E^u=1$, then $X/E^s$ is isomorphic to $E^u$, so a local stable manifold tangent to $E^s$ has codimension one.

Fix positive integers $a,b$ and a nonzero integer $c$. In positive integer solutions of $a{x}^n-b{y}^n=c$ with $n\ge 1$ and $Y=\max (x,y)>1$, the exponent $n$ is bounded effectively in terms of $a,b,c$.

When $x\ne y$ and the larger-variable power is sufficiently large, put $\Lambda =\log (a/b)+n\log (x/y)$. The equation makes $\Lambda$ nonzero and givesA Baker lower bound for a homogeneous linear form in logarithms gives $\log |\Lambda |\ge -C(a,b)\log Y\log (2n)$: the algebraic numbers are the rationals $a/b$ and $x/y$, the latter has height at most $Y$, and the coefficients are $1,n$. Dividing by $\log Y\ge \log 2$ yields $n\le C\log (2n)+O(1)$, which bounds $n$. If the larger-variable power is bounded, so is $n$. For $x=y>1$, the equation $(a-b)x^n=c$ gives the bound directly.

This is a bound on the exponent. Fixed exponents one or two can still admit infinitely many pairs $(x,y)$; for instance, $2x^2-y^2=1$ is a negative Pell equation after exchanging variables. The Thue theorem must not be applied to this quadratic case. The constant solution $x=y=1$, possible when $a-b=c$, is excluded by $Y>1$.

For $q_t-q_{xxx}=0$ on $x>0$, evaluate the global relation at $k=-ir$, where $\mathrm{Re}p>0$ and the principal cube root $r=p^{1/3}$ has positive real part. The time exponent becomes $-p$, giving the displayed Laplace transform identity. Thus the second-derivative trace is determined once the value and first-derivative traces are prescribed. The homogeneous transformed equation $u^{\prime \prime \prime }-pu=0$ has exactly two spatially decaying roots, so two independent admissible conditions are required. One condition leaves a free decaying mode; three independent conditions generally violate the global constraint.

For deterministic bounds $|M_k-M_{k-1}|\le c_k$, the conditional mean-zero increment has exponential moment at most $e^{{\theta }^2{c}_k^2/2}$. Iteration, the Markov inequality and optimization at $\theta =x/\sum c_k^2$ give each one-sided tail, and the union bound gives the displayed inequality. The factor two in the exponent denominator cannot be omitted: a single symmetric increment of size one has absolute deviation one with probability one.

When $U^{\prime }/r$ is constant and $U\ne c$, the axisymmetric inviscid pipe stability equation reduces to ${\varphi }^{\prime \prime }-{\varphi }^{\prime }/r-k^2\varphi =0$. Axis regularity selects $r{I}_1(kr)$ and impermeability gives $I_1(ka)=0$. These imaginary-wavenumber disturbances are spatially evanescent potential fields, not bounded whole-pipe temporal normal modes; their radial equation supplies no eigenvalue condition on $c$.