Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$

Depends on

• A linear map $L:\mathbb{R}^m\to\mathbb{R}^n$ in Euclidean coordinates
• The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$
• The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
• Finite sums and finite products, by recursion
• Laws of finite sums and finite products
• Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$, the parallelogram law and polarisation

Used by

• Lebesgue measure on ℝⁿ is invariant under every orthogonal linear map Corollary
• The finite-basis projector energy model supplies a C1 Hermitian response map Corollary
• The Jordan content of the parallelepiped spanned by the columns of a square real matrix is the absolute value of its determinant Corollary
• Generalized least-squares estimator for known covariance Definition
• Successive minima of a convex body Definition
• The Jacobian of a compact Riemann surface Definition
• A complex torus has a lattice of parabolic deck translations Example
• The complex torus as a Riemann surface Example
• The exponential map of a flat torus is not injective Example
• A coordinate scaling and a coordinate transposition send the unit cube to a set of measure equal to the absolute value of the determinant Lemma
• A Gaussian error contrast has the induced multivariate-normal law Lemma
• A parameterized C5 center manifold exists at a simple Hopf spectral split Lemma
• A shear sends the unit cube to a set of Lebesgue measure one Lemma
• A small residual gives a nearby root under a uniform inverse and Lipschitz bound in the maximum norm Lemma
• Area and L² derivative bounds for quasiconformal homeomorphisms Lemma
• Finite channel information maxima are attained Lemma
• Fundamental parallelotope and finite bounded intersections Lemma
• Invertibility and a norm bound for the pseudo-arclength bordered matrix Lemma
• Invertible linear substitutions preserve Schwartz space Lemma
• Linear statistics on Euclidean data are Borel-measurable Lemma
• Newton maps are uniform contractions near a point with invertible derivative Lemma
• Smooth orientation sign is the local integral homology multiplier Lemma
• The boundary of a convex set is Lebesgue null Lemma
• The Coxeter plane, ordered-root enumeration, and invertibility of the Coxeter action minus the identity Lemma
• The Dirac comb of a full-rank lattice transforms to the dual comb Lemma
• A holomorphic function of several variables is continuous and separately holomorphic Proposition
• Sums, products and nonvanishing quotients of holomorphic functions are holomorphic Proposition
• A conjugate difference quotient characterizes antiholomorphic maps Theorem
• A gap-controlled dimer mode gives a local saddle-translation error recurrence Theorem
• A linear endomorphism of ℝⁿ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant Theorem
• A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not Theorem
• A total derivative computes every directional derivative, and its matrix is the Jacobian Theorem
• An invertible linear map of ℝⁿ scales the Lebesgue measure of every Borel set by a positive constant depending only on the map Theorem
• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations Theorem
• Conditional Gaussian law of random effects Theorem
• Every affine hyperplane of ℝⁿ, and hence every proper linear subspace, is Lebesgue null Theorem
• Known-covariance BLUP and its prediction-error covariance Theorem
• Known-covariance GLS is BLUE Theorem
• Poisson summation for a full-rank lattice Theorem
• Poisson summation under two-sided polynomial decay Theorem

…and 4 more results.

Dependency tree · two levels

54 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis I, §8.3 (standard reference, not scraped)