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Examples:

* Reflection symmetry -> suffices to test odd and even functions separately.

* Translation invariance -> suffices to test individual plane waves (i.e., to inspect the Fourier multiplier symbol).

* Dilation invariance [if unitary] -> suffices to test homogeneous functions.

* Rotation invariance -> suffices to test the case of spherical harmonic behavior in angular variable (separation of variables). [This is the case for the MathOverflow post listed above.]

* etc.

(2/2)

Moving away from the Hilbert space context, some broader examples of this general philosophy:

* Invariance under linear combinations / convex combinations / algebraic operations -> suffices to check basis elements / extreme elements / generators

* Invariance under tensor products -> suffices to check one-dimensional/irreducible case

* Invariance under limits -> suffices to check a dense subclass

* Multiplicative structure (in analytic number theory) -> suffices to check prime powers

(3/2)

* Invariance under a symmetry -> suffices to test "normalized" objects (or restrict to a "section" of the symmetry). [I've referred to this trick in the past as "spending a symmetry".]

* Monotonicity -> suffices to check "sufficiently large/small" / limiting / extremal cases

* Estimate insensitive to multiplication by constants -> suffices to check "lacunary" values

* Estimate insensitive to logarithmic losses -> suffices to check simple functions / prove distributional estimates

(4/2)

* Estimate insensitive to perturbations by ε -> suffices to check a ε-net/verify a discretized analogue

* Invariance under "gluing" -> suffices to verify the "local" problem [esp. if there is some compactness]

* Gauge invariance -> suffices to work in a preferred gauge [a classic special case of the previous "normalization" example]

Perhaps the broader take-away is to always have situational awareness about all the (exact or approximate) invariances available for the problem at hand. (5/)

Can't resist adding some further examples:

* [Principle of induction] Preserved by successor -> suffices to test base case and/or limiting cases.

* Respects short exact sequences -> suffices to test "simple" objects (can also be used to separate into "solvable" and "semisimple" cases, "totally disconnected" and "connected" cases, "torsion" and "torsion-free" cases, etc.. Works particularly well with classification theorems, such as the classification of finite simple groups)

(6/)

* Insensitive to "pseudorandom"/"Gowers uniform" perturbations -> suffices to test "structured" objects (polynomials, nilsequences, "low rank" objects, etc.). [A core strategy of modern additive combinatorics.]

* Preserved by equivalence -> suffices to check one representative of each equivalence class. [Obvious and used everywhere, but perhaps worth stating explicitly.]

(7/)

* Invariance with respect to "small" / "lower order" / "asymptotically vanishing" perturbations (or elements of an ideal) -> suffices to check in domains where these perturbations have been "quotiented out", for instance by working in some asymptotic limit.

* Invariance with respect to "base change" or change of underlying field -> suffices to check a favorable base (e.g., a field that is finite, characteristic zero, or algebraically closed). [Combines well with model theory tools.]

(8/)

* Estimate insensitive to multiplication by constants -> suffices to control individual terms of estimate (if boundedly many terms, or if one gains summable decay in the index of the term). [A fundamental tool in harmonic analysis and related fields, often considered too trivial to state explicitly.]

* Insensitive to specific relations (or other features) of domain / preserved by quotients -> suffices to verify the "universal" domain (if one exists), or work in a more abstract setting

(9/)

* Preserved by analytic continuation -> suffices to check some indiscrete range of complex parameter / work with formal power series expansions in that parameter

* Is closed under "induction on scales" / "concatenation" / "semigroup composition" -> suffices to check an infinitesimal (or "single scale", or "differentiated") version [but often one cannot afford to "lose constants" after applying this reduction]

(10/)

* Preserved by "disjoint union" -> suffices to check "connected" spaces, "spanning" sets, "cycle" permutations, or "ergodic" systems

* Preserved by "asymptotically separated superposition" and by "dispersed perturbations" -> suffices to check "almost periodic/compact modulo symmetry" objects. [This is the philosophy behind the concentration-compactness approach to PDE, especially its more modern applications in critical dispersive PDE that are based on profile decompositions.]

(11/)

@theking @tao one minor comment - sometimes these things require doing something "unnatural" from a category theory perspective. Like it can be convenient to pick a basis to prove results in linear algebra, but there is no natural choice of basis.