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For a linear map $T:V\to W$ between finite-dimensional inner product spaces,The first equality follows directly from the defining identity for the adjoint operator; the second follows by taking orthogonal complements. In an infinite-dimensional Hilbert space, the second equality generally requires closure of the image.

The induced trace norm of a linear map $T$ between matrix spaces isIt measures the largest trace-norm amplification available without an auxiliary tensor factor.

An intertwiner between group representations $\rho$ on $V$ and $\sigma$ on $W$ is a linear map $A:V\to W$ satisfying $A\rho (x)=\sigma (x)A$ for every group element. The Schur lemma controls these maps between irreducible representations.

An invariant subspace of a linear map $A:V\to V$ is a vector subspace $W\subseteq V$ satisfying $A(W)\subseteq W$. For a group representation, invariance means $\rho (g)W\subseteq W$ for every represented group element $g$.

An invertible linear map is a linear bijection; its inverse is also linear.

An isomorphism of Banach spaces is a bijective bounded linear map whose inverse is bounded. The bounded inverse theorem makes boundedness of the inverse automatic.

A Lie algebra homomorphism is a linear map $f$ satisfying $f([x,y])=[f(x),f(y)]$.

A Lie algebra isomorphism is a bijective linear map $\phi :\mathfrak{g}\to \mathfrak{h}$ satisfying $\phi ([x,y])=[\phi (x),\phi (y)]$. Its inverse automatically preserves the Lie bracket.

A linear isomorphism is a bijective linear map. Its inverse is also linear.

A linear operator is a linear map from a vector space to itself.

Linear operator theory studies linear maps, their spectra, invariant subspaces, and normal forms.

A matrix represents a linear map after bases have been chosen for its domain and codomain.

A topological vector space is metrizable when its topology is induced by a metric. At the origin it then has a countable neighborhood basis, so continuity of a linear map can be tested on sequences.

A module isomorphism is a bijection of modules over the same ring preserving addition and scalar multiplication. Its inverse preserves these operations too. When a linear operator $T$ gives a vector space an $F[t]$-module structure by $t\cdot v=T(v)$, a module isomorphism is exactly an invertible linear map intertwining the two operators. This translates classification of modules into matrix similarity and rational canonical form.

A multilinear map $F:V_1\times \cdots \times V_r\to W$ is a map between vector spaces that is linear in each argument when all other arguments are fixed. For example the evaluation $(\omega ,v)\mapsto \omega (v)$ is bilinear, and the determinant is multilinear in its columns. The universal property of a tensor product replaces this separate linearity by a single linear map from $V_1\otimes \cdots \otimes V_r$.

For at least two distinct knots $x_1<\cdots <x_n$ and values $v_i$, the unique natural cubic spline matching those values and linear beyond $x_1,x_n$. Its interpolation map is a linear map by uniqueness. Over a larger domain $[a,b]$, the two exterior pieces remain linear. With only one knot, arbitrary slopes of affine functions make uniqueness false.

A linear map $T:V\to V$ is nilpotent if $T^k=0$ for some positive integer $k$.

A continuous surjective linear map between Fréchet spaces is an open map. The Baire category theorem first shows that the closure of the image of every convex balanced zero-neighbourhood is a zero-neighbourhood: the target is the countable union of scalar multiples of this image, and an interior point of its closure can be translated to zero using convexity and symmetry.

To remove the closure, fix a domain zero-neighbourhood $U$ and choose convex balanced zero-neighbourhoods $U_n$ so that every series ${\sum }_n{u}_n$, $u_n\in U_n$, converges to a point of $U$. With an increasing defining sequence of seminorms $p_j$, one can ensure $p_j(u_n)<2^{-n}$ for $j\le n$ and make the finitely many seminorms defining $U$ summable within its bounds. Completeness gives convergence. Choose target zero-neighbourhoods $V_n\subset \overline{A(U_n)}$ shrinking to zero. For $y\in V_1$, choose $u_n\in U_n$ successively so that $y-A{\sum }_{j\le n}{u}_j\in V_{n+1}$. Density permits each correction. Continuity then gives $y=A{\sum }_n{u}_n\in A(U)$, proving openness.

For Fréchet spaces, the two continuity conditions for a linear map are equivalent. Original continuity makes every target continuous linear functional compose to a source continuous linear functional, proving weak continuity. Conversely, weak continuity makes these compositions continuous in the original source topology. They separate target points, so limits in the graph of a linear operator remain in the graph. The closed graph theorem for Fréchet spaces finishes the proof.

Without the hypotheses, equal continuous dual spaces do not imply equal topologies. The identity from an infinite-dimensional Hilbert space with its weak topology to its norm topology is weak-to-weak continuous but not continuous in those original topologies.