Functional analysis

Around 1900 Ivar Fredholm in Stockholm studied integral equations $u(x) - \lambda \int_0^1 K(x, y) u(y) \, dy = f(x)$, which arise from boundary value problems and potential theory. He noticed that they behave like systems of linear equations: either the homogeneous equation has a non-zero solution, or the inhomogeneous equation has exactly one solution for every $f$. David Hilbert and his students in Göttingen then built a theory of "infinitely many unknowns", expanding functions in orthogonal series and diagonalising symmetric kernels — the infinite-dimensional version of diagonalising a symmetric matrix. Stefan Banach and the Lwów school in the 1920s abstracted the setting into complete normed spaces.

The same structures run through modern mathematics: the state space of quantum mechanics is a Hilbert space, the natural home of a weak solution of a PDE is a Sobolev space, the space $L^1$ of probability densities is a Banach space, and optimisation in infinite dimensions needs duality. Functional analysis is linear algebra plus topology, and its lesson is double: much of finite-dimensional linear algebra survives, provided one insists on completeness and continuity, and some of it fails in instructive ways.

In $\R^n$ every linear map is continuous, all norms are equivalent, the closed unit ball is compact, and a symmetric matrix has an orthonormal basis of eigenvectors. In infinite dimensions:

• Continuity is no longer automatic. Differentiation $f \mapsto f'$ is linear, but $\sin(nx)$ has sup norm $1$ while its derivative has sup norm $n$. So we must restrict attention to bounded operators, those satisfying $\|Tx\| \le C\|x\|$.
• Norms can be inequivalent. On $C[0, 1]$, $\|x^n\|_\infty = 1$ but $\|x^n\|_1 = 1/(n+1)$; the two norms give different notions of convergence, and only the first makes $C[0, 1]$ complete.
• The unit ball is not compact. In $\ell^2$ the unit vectors $e_n$ satisfy $\|e_n - e_m\| = \sqrt 2$ for $n \ne m$, so no subsequence converges. Compactness arguments must use weaker topologies.
• Eigenvalues can vanish. The right shift $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$ on $\ell^2$ has no eigenvalues at all.

What survives is carried by completeness. Through the Baire category theorem, completeness yields three "automatic" theorems: pointwise bounded families of operators are uniformly bounded, bijective bounded operators have bounded inverses, and operators with closed graphs are bounded. Through the axiom of choice, the Hahn–Banach theorem supplies enough continuous linear functionals to separate points. In a Hilbert space, the inner product adds geometry — orthogonal projection, Pythagoras — and the Riesz representation theorem identifies the space with its dual. And for compact self-adjoint operators the finite-dimensional spectral theorem survives almost unchanged.

Every separable Hilbert space has a countable orthonormal basis (Gram–Schmidt applied to a dense sequence), so every infinite-dimensional separable Hilbert space is isometrically isomorphic to $\ell^2$ by $x \mapsto (\langle x, e_n \rangle)_n$. In particular $L^2(-\pi, \pi) \cong \ell^2(\Z)$ by Fourier coefficients: Parseval's identity for Fourier series is (d).

The engine of the next three theorems is the Baire category theorem: in a complete metric space, a countable intersection of dense open sets is dense; equivalently, a complete metric space is not a countable union of nowhere dense closed sets. (Proof: given a ball $B_0$ and dense open $U_1, U_2, \ldots$, choose closed balls $\overline{B_n} \subseteq B_{n-1} \cap U_n$ with radii $\to 0$; their centres form a Cauchy sequence whose limit lies in $B_0 \cap \bigcap U_n$. See point-set topology.)

Completeness is essential. On the space $c_{00}$ of finitely supported sequences with the sup norm, $T_n(x) = n x_n$ satisfies $\sup_n |T_n x| < \infty$ for each $x$ (only finitely many $x_n \ne 0$), but $\|T_n\| = n$. A famous application: there is a continuous $2\pi$-periodic function whose Fourier series diverges at $0$, because the partial-sum functionals $f \mapsto S_N f(0)$ on $C(\mathbb T)$ have norms equal to the Lebesgue constants $\|D_N\|_{L^1} \sim \frac{4}{\pi^2} \log N \to \infty$.

About the proof. For (1), $Y = \bigcup_n \overline{T(B(0, n))}$, so by Baire some $\overline{T(B(0, n))}$ has interior, and by linearity $\overline{T(B(0, 1))} \supseteq B(0, \delta)$ for some $\delta > 0$. A successive approximation argument, summing a series that converges by completeness of $X$, removes the closure: $T(B(0, 2)) \supseteq B(0, \delta)$. See Brezis, Theorem 2.6. (2) follows from (1): the graph $\Gamma$ is a closed subspace of $X \times Y$, hence Banach, and the projection $\Gamma \to X$, $(x, Tx) \mapsto x$, is a bounded bijection; its inverse $x \mapsto (x, Tx)$ is therefore bounded, and so is $T$.

In practice the closed graph theorem says: to prove that $T$ is bounded, it suffices to show that $x_n \to x$ and $Tx_n \to y$ imply $y = Tx$. For example, a symmetric operator defined on all of a Hilbert space ($\langle Tx, y \rangle = \langle x, Ty \rangle$) is automatically bounded (Hellinger–Toeplitz). This is why the unbounded operators of quantum mechanics, such as momentum $-i \, d/dx$, are only defined on dense subspaces.

About the proof. One extends $f$ by one dimension at a time — the inequality $f \le p$ leaves an interval of admissible values for $F(x_0)$ — and Zorn's lemma, applied to the partially ordered set of dominated extensions, produces a maximal one, which must be defined everywhere. The complex case follows by applying the real case to $\operatorname{Re} f$. Consequences: $X^*$ separates points; $\|x\| = \max_{\|\ell\| \le 1} |\ell(x)|$, so $X$ embeds isometrically in its bidual $X^{**}$; and (in its geometric form) disjoint convex sets, one open, can be separated by a closed hyperplane — the foundation of convex optimisation and duality.

Compactness returns in a weaker topology. Banach–Alaoglu: the closed unit ball of $X^*$ is weak-* compact. In a Hilbert space (indeed in any reflexive space) every bounded sequence has a weakly convergent subsequence. The orthonormal sequence above converges weakly to $0$: $\langle e_n, y \rangle \to 0$ for every $y$, by Bessel's inequality. Weak compactness is the standard tool for producing solutions of variational problems and PDEs: minimise, extract a weakly convergent minimising sequence, and use lower semicontinuity of the norm, $\|x\| \le \liminf \|x_n\|$ when $x_n \rightharpoonup x$.

Let $H$ be a Hilbert space and $T \in \mathcal B(H)$ self-adjoint. Then $\langle Tx, x \rangle = \langle x, Tx \rangle = \overline{\langle Tx, x \rangle}$ is real, eigenvalues are real, and eigenvectors for different eigenvalues are orthogonal: $\lambda \langle x, y \rangle = \langle Tx, y \rangle = \langle x, Ty \rangle = \mu \langle x, y \rangle$. If $T$ leaves a subspace $M$ invariant, it also leaves $M^\perp$ invariant: for $y \in M^\perp$ and $x \in M$, $\langle Ty, x \rangle = \langle y, Tx \rangle = 0$.

Applied to the Green's operator $G$ of $-u'' = f$ on $(0, 1)$, the theorem produces the sine basis $\sqrt 2 \sin(k\pi x)$ of $L^2(0, 1)$ — Fourier series as a spectral theorem. Applied to $I - \lambda K$ with $K$ compact and self-adjoint, it gives the Fredholm alternative: for $\lambda \ne 0$, either $\lambda^{-1}$ is an eigenvalue of $K$ or $I - \lambda K$ is invertible (the Riesz–Schauder theory extends this to every compact $K$). For bounded self-adjoint operators that are not compact (multiplication by $x$ on $L^2(0, 1)$, which has no eigenvalues at all), the spectral theorem needs projection-valued measures: von Neumann (1929) and Stone (1932), extending it to unbounded operators for quantum mechanics.

Mathlib distinguishes E →ₗ[ℝ] F (linear maps) from E →L[ℝ] F (continuous linear maps) — a type-level reminder that continuity is extra structure in infinite dimensions. The first half of "bounded ⇔ continuous", proved directly:

/-- **Bounded linear maps are continuous.** If `‖T x‖ ≤ C ‖x‖` for all `x`, then
`‖T y − T x‖ = ‖T (y − x)‖ ≤ C ‖y − x‖`, so `T` is (Lipschitz) continuous: given `ε > 0`,
take `δ = ε / (|C| + 1)`. (Mathlib: `LinearMap.continuous_of_bound`; the converse, that a
continuous linear map is bounded, is `ContinuousLinearMap.le_opNorm`.) -/
theorem continuous_of_bound {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [NormedAddCommGroup F] [NormedSpace ℝ F] (T : E →ₗ[ℝ] F) (C : ℝ)
    (h : ∀ x, ‖T x‖ ≤ C * ‖x‖) : Continuous T := by
  rw [Metric.continuous_iff]
  intro x ε hε
  have hC : 0 < |C| + 1 := by positivity
  refine ⟨ε / (|C| + 1), by positivity, fun y hy => ?_⟩
  rw [dist_eq_norm] at hy ⊢
  rw [← map_sub]
  calc ‖T (y - x)‖ ≤ C * ‖y - x‖ := h (y - x)
    _ ≤ (|C| + 1) * ‖y - x‖ := by
        apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
        linarith [le_abs_self C]
    _ < (|C| + 1) * (ε / (|C| + 1)) := mul_lt_mul_of_pos_left hy hC
    _ = ε := by field_simp

The Riesz representation theorem is Mathlib's isometric conjugate-linear equivalence InnerProductSpace.toDual : E ≃ₗᵢ⋆[𝕜] StrongDual 𝕜 E. Note that Mathlib's inner product is conjugate-linear in the first argument (the physicists' convention), so the representing vector goes on the left. Uniqueness, proved as in the lesson:

/-- **Riesz representation theorem** (Fréchet–Riesz). On a Hilbert space `E` over `𝕜 = ℝ` or `ℂ`,
every bounded linear functional `ℓ` is an inner product with a fixed vector:
`ℓ x = ⟪y, x⟫` for some `y`. (Mathlib's inner product is conjugate-linear in the *first*
slot.) This applies Mathlib's isometric equivalence `InnerProductSpace.toDual`. -/
theorem riesz_representation {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E]
    [InnerProductSpace 𝕜 E] [CompleteSpace E] (ℓ : StrongDual 𝕜 E) :
    ∃ y : E, ∀ x, ℓ x = ⟪y, x⟫_𝕜 :=
  ⟨(InnerProductSpace.toDual 𝕜 E).symm ℓ, fun _ => InnerProductSpace.toDual_symm_apply.symm⟩

/-- **Uniqueness of the representing vector.** If `⟪y₁, x⟫ = ⟪y₂, x⟫` for every `x`, then
`y₁ = y₂`: take `x = y₁ − y₂` to get `⟪y₁ − y₂, y₁ − y₂⟫ = 0`, and an inner product is
definite. (This does not need completeness; existence does.) -/
theorem riesz_unique {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
    {y₁ y₂ : E} (h : ∀ x, ⟪y₁, x⟫_𝕜 = ⟪y₂, x⟫_𝕜) : y₁ = y₂ := by
  have h0 : ⟪y₁ - y₂, y₁ - y₂⟫_𝕜 = 0 := by
    rw [inner_sub_left, h (y₁ - y₂), sub_self]
  exact sub_eq_zero.mp (inner_self_eq_zero.mp h0)

The three Baire-category theorems and Hahn–Banach, all library theorems:

/-- **Uniform boundedness principle** (Banach–Steinhaus). A family of bounded operators on a
Banach space that is bounded at each point is bounded in operator norm. Proved in Mathlib
(`banach_steinhaus`) from the Baire category theorem, as in the lesson. -/
theorem uniform_boundedness {ι E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] {T : ι → E →L[ℝ] F}
    (h : ∀ x, ∃ C, ∀ i, ‖T i x‖ ≤ C) : ∃ C', ∀ i, ‖T i‖ ≤ C' :=
  banach_steinhaus h

/-- **Open mapping theorem** (Banach–Schauder). A surjective bounded operator between Banach
spaces maps open sets to open sets. This is Mathlib's `ContinuousLinearMap.isOpenMap`. -/
theorem open_mapping {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
    [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] (T : E →L[ℝ] F)
    (hT : Function.Surjective T) : IsOpenMap T :=
  T.isOpenMap hT

/-- **Closed graph theorem.** A linear map between Banach spaces whose graph is closed in
`E × F` is continuous. This is Mathlib's `LinearMap.continuous_of_isClosed_graph`. -/
theorem closed_graph {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
    [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] (T : E →ₗ[ℝ] F)
    (hT : IsClosed (T.graph : Set (E × F))) : Continuous T :=
  T.continuous_of_isClosed_graph hT

/-- **Hahn–Banach theorem** (normed version). A bounded linear functional on a subspace extends
to the whole space *with the same norm*. Mathlib proves it (`exists_extension_norm_eq`) from
the sublinear-functional version, which uses Zorn's lemma. -/
theorem hahn_banach {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (p : Subspace ℝ E)
    (f : StrongDual ℝ p) : ∃ g : StrongDual ℝ E, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ :=
  exists_extension_norm_eq p f

The non-compactness of the unit ball, proved from scratch from an orthonormal sequence (the lesson's proof), next to Mathlib's converse in arbitrary normed spaces:

/-- Distinct members of an orthonormal family are at distance `√2`:
`‖eᵢ − eⱼ‖² = ‖eᵢ‖² − 2⟪eᵢ, eⱼ⟫ + ‖eⱼ‖² = 1 − 0 + 1 = 2` (Pythagoras). -/
theorem orthonormal_dist_sq {E ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    {e : ι → E} (he : Orthonormal ℝ e) {i j : ι} (hij : i ≠ j) : ‖e i - e j‖ ^ 2 = 2 := by
  rw [norm_sub_sq_real, he.1 i, he.1 j, he.2 hij]
  norm_num

/-- **The closed unit ball of an infinite-dimensional Hilbert space is not compact.** If `E` has
an orthonormal sequence `e₀, e₁, …` (every infinite-dimensional Hilbert space does, by
Gram–Schmidt), the `eₙ` lie in the closed unit ball but no subsequence is Cauchy, because
distinct terms are `√2 > 1` apart. A compact metric space would contain a convergent, hence
Cauchy, subsequence. -/
theorem closedBall_not_compact {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    {e : ℕ → E} (he : Orthonormal ℝ e) : ¬ IsCompact (closedBall (0 : E) 1) := by
  intro hK
  have hmem : ∀ n, e n ∈ closedBall (0 : E) 1 := fun n => by
    rw [mem_closedBall, dist_zero_right, he.1 n]
  obtain ⟨a, -, φ, hφ, hlim⟩ := hK.tendsto_subseq hmem
  -- A convergent sequence is Cauchy: eventually all terms are within `1` of each other.
  obtain ⟨N, hN⟩ := Metric.cauchySeq_iff'.1 hlim.cauchySeq 1 one_pos
  have hlt : dist (e (φ (N + 1))) (e (φ N)) < 1 := hN (N + 1) (Nat.le_succ N)
  have hne : φ (N + 1) ≠ φ N := (hφ (Nat.lt_succ_self N)).ne'
  have h2 := orthonormal_dist_sq he hne
  rw [dist_eq_norm] at hlt
  nlinarith [norm_nonneg (e (φ (N + 1)) - e (φ N))]

/-- **F. Riesz's theorem.** Conversely, in any normed space, a compact closed ball of positive
radius forces finite dimension. This is Mathlib's `FiniteDimensional.of_isCompact_closedBall`,
proved from Riesz's lemma. -/
theorem riesz_compact_ball {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {r : ℝ}
    (hr : 0 < r) (h : IsCompact (closedBall (0 : E) r)) : FiniteDimensional ℝ E :=
  FiniteDimensional.of_isCompact_closedBall ℝ hr h

Mathlib's spectral theorem for compact self-adjoint operators is stated as: the orthogonal complement of the span of all eigenspaces is zero, and eigenspaces for non-zero eigenvalues are finite-dimensional. Together with the orthogonality of eigenspaces this is the theorem proved above.

/-- **Spectral theorem for compact self-adjoint operators**, completeness part. For a compact
symmetric operator `T` on a Hilbert space, the eigenspaces span a dense subspace: the orthogonal
complement of their span is `0`. Choosing an orthonormal basis of each (finite-dimensional, for
non-zero eigenvalues) eigenspace gives an orthonormal basis of eigenvectors. This is Mathlib's
`ContinuousLinearMap.orthogonalComplement_iSup_eigenspaces_eq_bot`. -/
theorem spectral_compact_symmetric {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E]
    [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : IsCompactOperator T)
    (hT' : (T : E →ₗ[𝕜] E).IsSymmetric) :
    (⨆ μ, Module.End.eigenspace (T : Module.End 𝕜 E) μ)ᗮ = ⊥ :=
  T.orthogonalComplement_iSup_eigenspaces_eq_bot hT hT'

/-- The eigenspace of a compact operator for a non-zero eigenvalue is finite-dimensional
(its unit ball is the image of a bounded set under `μ⁻¹ T`, hence compact). This is Mathlib's
`ContinuousLinearMap.finite_dimensional_eigenspace`. -/
theorem eigenspace_finite_dimensional {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E]
    [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : IsCompactOperator T)
    (μ : 𝕜) (hμ : μ ≠ 0) : FiniteDimensional 𝕜 (Module.End.eigenspace T.toLinearMap μ) :=
  T.finite_dimensional_eigenspace hT μ hμ

Not formalised here: the projection theorem (Mathlib: exists_norm_eq_iInf_of_complete_convex), Bessel and Parseval (Mathlib: Orthonormal.tsum_inner_products_le and HilbertBasis), our proof of the spectral theorem (we cite Mathlib's, which goes through the spectral radius), weak topologies and Banach–Alaoglu (Mathlib: WeakDual.isCompact_polar), and the spectral theorem for non-compact operators.

The experiment discretises the Green's operator $G$ of $-u''$ on $(0, 1)$ by a $200 \times 200$ symmetric matrix (the midpoint rule) and compares its eigenvalues with the exact values $1/(k\pi)^2$. The matrix eigenvectors are orthonormal, the first matches $\sqrt 2 \sin(\pi x)$, and the count of eigenvalues above each threshold matches the count for the operator: they pile up only at $0$, as the spectral theorem requires. Two exact computations then show the warnings of the intuition section: differentiation multiplies the sup norm of $\sin(nx)$ by $n$, and $\|x^n\|_\infty / \|x^n\|_1 = n + 1$.

"""The spectrum of a compact self-adjoint operator, seen through a matrix.

The operator (Gf)(x) = integral of G(x, y) f(y) dy on L^2(0, 1), with kernel
G(x, y) = min(x, y) (1 - max(x, y)), inverts -u'' = f, u(0) = u(1) = 0. It is
compact and self-adjoint, with eigenvalues 1/(k pi)^2 and eigenfunctions
sqrt(2) sin(k pi x). We discretise it by the midpoint rule (an N x N
symmetric matrix), compare its eigenvalues with the exact ones, and count how
they pile up at 0. Then two infinite-dimensional warnings: differentiation is
unbounded, and the sup and L^1 norms are not equivalent.
"""

import math
from fractions import Fraction

import numpy as np

def green_matrix(n):
    """Midpoint-rule matrix (1/n) G(x_i, x_j) with x_i = (i + 1/2)/n."""
    if n < 1:
        raise ValueError("n must be positive")
    x = (np.arange(n) + 0.5) / n
    big_x, big_y = np.meshgrid(x, x, indexing="ij")
    kernel = np.minimum(big_x, big_y) * (1 - np.maximum(big_x, big_y))
    return kernel / n

def eigen_decreasing(matrix):
    """Eigenvalues (decreasing) and orthonormal eigenvectors of a symmetric
    matrix."""
    values, vectors = np.linalg.eigh(matrix)
    return values[::-1], vectors[:, ::-1]

def exact_eigenvalue(k):
    """The k-th eigenvalue 1/(k pi)^2 of the Green's operator."""
    return 1 / (k * math.pi) ** 2

def count_above(values, threshold):
    """How many eigenvalues exceed the threshold."""
    return int(np.sum(np.asarray(values) > threshold))

def exact_count_above(threshold):
    """Number of k >= 1 with 1/(k pi)^2 > threshold."""
    k = 0
    while exact_eigenvalue(k + 1) > threshold:
        k += 1
    return k

def derivative_ratio(n, samples=4000):
    """max |f'| / max |f| for f(x) = sin(n x) on [0, 2 pi], sampled."""
    xs = np.linspace(0, 2 * math.pi, samples)
    return float(np.max(np.abs(n * np.cos(n * xs))) /
                 np.max(np.abs(np.sin(n * xs))))

def power_norms(n):
    """Exact (sup norm, L^1 norm) of x^n on [0, 1]."""
    return Fraction(1), Fraction(1, n + 1)

def main():
    n = 200
    values, vectors = eigen_decreasing(green_matrix(n))
    print(f"Green's operator of -u'' on (0, 1), midpoint matrix with N = {n}")
    print(f"{'k':>3} {'matrix':>12} {'exact 1/(k pi)^2':>18} {'rel. error':>11}")
    for k in range(1, 7):
        exact = exact_eigenvalue(k)
        err = abs(values[k - 1] - exact) / exact
        print(f"{k:>3} {values[k - 1]:>12.8f} {exact:>18.8f} {err:>11.1e}")
    orth = np.max(np.abs(vectors.T @ vectors - np.eye(n)))
    print(f"max |V^T V - I| = {orth:.1e} (orthonormal eigenvectors)")
    first = vectors[:, 0] * math.sqrt(n)
    sine = math.sqrt(2) * np.sin(math.pi * (np.arange(n) + 0.5) / n)
    first = first if first @ sine > 0 else -first
    print(f"first eigenvector vs sqrt(2) sin(pi x): max gap "
          f"{np.max(np.abs(first - sine)):.1e}")
    for t in [1e-1, 1e-2, 1e-3]:
        print(f"eigenvalues > {t:g}: matrix {count_above(values, t)}, "
              f"exact {exact_count_above(t)}")
    print(f"smallest matrix eigenvalue: {values[-1]:.2e} (they pile up at 0)")
    print()
    print("Differentiation is unbounded: f = sin(n x), ratio max|f'| / max|f|")
    print("  ", [round(derivative_ratio(m), 3) for m in [1, 10, 100]])
    print("Norms are not equivalent: x^n on [0, 1], (sup norm, L^1 norm)")
    print("  ", [tuple(str(v) for v in power_norms(m)) for m in [1, 10, 100]])
    print()
    print("Epistemic status: the eigenvalue table is a numerical approximation")
    print("of one operator (midpoint rule, floating point), within about 2e-6")
    print("of the exact eigenvalues; it proves nothing in general. The derivative")
    print("ratios are sampled, the norms of x^n exact. That no constant bounds")
    print("either ratio follows because n and n + 1 are unbounded.")

if __name__ == "__main__":
    main()

This is numerical evidence about one operator. Why the matrix eigenvalues approximate the operator's is itself a theorem (convergence of the Nyström method for compact operators); the spectral theorem is what guarantees there is a discrete spectrum to approximate.

In two dimensions the spectral theorem is visible: a symmetric matrix stretches the plane along two perpendicular invariant lines, its eigenvectors. Try other symmetric matrices, then a non-symmetric shear, whose single invariant line is not accompanied by a perpendicular one.

In infinite dimensions, boundedness can fail. Move the slider: $\sin(nx)$ stays between $-1$ and $1$, but its derivative $n\cos(nx)$ grows without bound. No constant $C$ gives $\|f'\|_\infty \le C\|f\|_\infty$ on all of $C^1$.