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Let $b_i=N{e}_i$ be the interpolating cardinal basis and set ${\Gamma }_{ij}={\int }_a^b{b}_i^{\prime \prime }{b}_j^{\prime \prime }dx$. Linearity gives $J(N\mathbf{v})={\mathbf{v}}^T\Gamma \mathbf{v}$, and this Gram matrix is a positive semidefinite matrix. Its null space consists precisely of vectors $(\alpha +\beta x_i{)}_i$, by the null-space statement for the second derivative roughness penalty. Thus its rank is $n-2$, including the zero matrix when $n=2$.

The standard basis of the vector space ${\mathbb{R}}^n$ consists of the vectors $e_i$ whose $i$th coordinate is one and whose other coordinates are zero. Every vector $x$ has the unique expression $x={\sum }_i{x}_i{e}_i$ in this basis.

A straight line in Euclidean space is the set $L=\{x_0+tv:t\in \mathbb{R}\}$, for a fixed point $x_0$ and nonzero direction vector $v$. Two distinct points determine one such line.

For a real vector $x$ with support of a vector $S$, this certificate is a vector $h$ whose measurements under the adjoint operator match the active sign function values and lie strictly between minus one and one on the inactive coordinates. If $A_S$ is injective, such a certificate proves that $x$ is the unique basis pursuit minimizer. For a nonzero $v\in \ker A$, injectivity ensures $v_{S^c}\ne 0$, and $0=⟨A^{\ast }h,v⟩$ implies the fixed-sign null space condition. The strict inequality is interpreted coordinatewise if $S^c$ is empty. The condition with the sign function written here is for real variables; complex basis pursuit uses the unit phases of active coordinates instead.

A real symmetric matrix whose quadratic form is positive at every nonzero vector in the nonnegative orthant. By compactness of the nonnegative unit sphere, this is equivalent to a positive uniform lower bound $x^{T}Ax\ge \delta \Vert x{\Vert }_2^2$ on that orthant. It need not be a positive-definite matrix.

For positive-semidefinite kernels $k_j$ with Reproducing-kernel Hilbert spaces ${\mathcal{H}}_j$, the Reproducing-kernel Hilbert space of $k={\sum }_j{k}_j$ consists of sums $f={\sum }_j{f}_j$, withThe minimum is attained uniquely. Indeed, in the Hilbert space direct sum $E={⨁}_j{\mathcal{H}}_j$, the tuples summing to zero form a closed subspace $N$: they are the intersection, over inputs $x$, of the kernels of the continuous maps $(f_j)\mapsto {\sum }_j{f}_j(x)$. Every affine fibre has a unique representative in $N^{\perp }$ by orthogonal decomposition by a closed subspace. The vector $(k_j(\cdot ,x){)}_j$ lies in $N^{\perp }$ and represents evaluation in the quotient; its evaluation at $y$ is ${\sum }_j{k}_j(y,x)$, identifying the positive-semidefinite kernel with $k$.

If contraction of an array with every symmetric second-rank tensor is a vector in every orthonormal frame, its part symmetric in the contracted slots obeys the rank-three tensor transformation law. The difference between its proposed transformation and actual components is symmetric and contracts to zero against every symmetric matrix, so it vanishes. The antisymmetric part is invisible, by the blindness of symmetric contraction tests to antisymmetric arrays, and need not be tensorial.

A multilinear operation on vector fields is tensorial when it is linear over smooth functions in every argument. Its value at a point then depends only on the argument vectors at that point. To see this, use a smooth cutoff function to reduce to local fields, expand them in a local frame, and apply linearity over smooth functions to their coefficients. A smoothly valued tensorial operation therefore defines a tensor field. The curvature of an affine connection is tensorial, whereas a covariant derivative differentiates a scalar coefficient in its second argument and is not tensorial there.

For an absolutely one-homogeneous functional $J=\mathrm{TV}$ on a real Hilbert space, a subgradient $q$ at $u$ is characterized by $⟨q,v⟩\le J(v)$ for every $v$ and $⟨q,u⟩=J(u)$. A bounded vector field with $|z|\le 1$ and $q=\mathrm{div}z$ proves the inequality through the dual definition, provided cutoff approximation is valid in the ambient space. Equality is called a calibration. It certifies a global minimizer of total variation denoising, rather than only a minimizer within a chosen ansatz.

A unit vector is a vector whose norm is one.

A vector $k$-coloring, for $k>1$, assigns a unit vector $v_i$ to each vertex of a graph, such that $⟨v_i,v_j⟩\le -1/(k-1)$ on every edge. A graph colouring with $k$ colors gives a vector $k$-coloring by placing the colors at the vertices of a regular simplex. The Gram matrix of these vectors allows semidefinite programming to search for such a representation.

A vector order parameter has several components transforming together as a vector under spatial rotations. A polar order parameter is an example. Its functional derivative $\delta F/\delta \mathbf{p}$ has one component for each component of the order parameter.

A vector space consists of vectors that may be added and multiplied by scalars, subject to the vector-space axioms.

The weak gradient is the vector of first-order weak derivatives of a function. Its $i$th component is specified by $\int u{\partial }_i\phi =-\int ({\partial }_{i}u)\phi$ for compactly supported test functions. For a differentiable function with appropriate local integrability it agrees with the classical gradient.