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The ground-state energy $E_0$ is the energy eigenvalue attained by a ground state. For a self-adjoint operator $H$ bounded below, distinguish this attained value from the always-defined spectral bottom $E_{\ast }=\inf \sigma (H)$. When a ground state exists, $E_0=E_{\ast }$. Some usage calls $E_{\ast }$ the ground-state energy even when it is unattained; here an unattained bottom is explicitly identified as an infimum. The quantum variational principle statesApproaching this infimum with trial wavefunctions does not ensure that a minimizing wavefunction exists. ^[ref].

For the whole-line free particle with mass $m>0$,Its spectrum is continuous and contains no energy eigenvalues represented by normalizable wavefunctions. The Fourier transform makes $H$ multiplication by ${\hslash }^2{k}^2/(2m)$; a fixed energy restricts $k$ to a set of zero length, on which a nonzero square-integrable function cannot be supported. ^[ref]. An explicit family of normalized Gaussian wave packets demonstrates how the bottom is approached:The formula follows from ${\psi }_s^{\prime }=-x{\psi }_s/(2s^2)$, integration by parts, and $⟨x^2{⟩}_s=s^2$. Every finite width gives positive kinetic energy. At zero energy, $H\psi =0$ would require an affine function on the whole line, which is never a nonzero normalizable wavefunction. Spreading a Gaussian wave packet indefinitely produces an unattained infimum, rather than a zero-energy ground state.

An additive choice of energy origin changes the numerical ground-state energy without changing the ground-state subspace. For $H^{\prime }=H+CI$, every energy eigenvalue becomes $E^{\prime }=E+C$, while all differences $E_j-E_i$ stay fixed. Time evolution gains only the global phase $e^{-iCt/\hslash }$. This makes excitation costs and spectroscopic differences more intrinsic than the isolated number $E_0$.

Hausdorff measure measures the geometric size of a set using covers by increasingly small pieces. The exponent $s$ chooses the measuring scale: $s=1$ is length-like, $s=2$ is area-like, and noninteger exponents can measure fractals. Unlike a count of occupied grid cells, the construction allows pieces of different sizes and considers every admissible cover.

For $s>0$ and a subset $E$ of a metric space, use the diameter-cost conventionThe first quantity is scale-restricted Hausdorff content. Shrinking $\delta$ removes options from the infimum, so these contents increase to a limit, possibly infinite. The resulting outer measure restricts to a measure on the Borel sets. For $s=0$, counting each nonempty covering piece with cost one gives counting measure.

A different common convention multiplies these quantities by $c_s={\pi }^{s/2}/(2^s\Gamma (1+s/2))$, using the Gamma function. At integer $s$, that normalization agrees with ordinary Euclidean $s$-dimensional volume. Here $c_1=1$, so both conventions make ${\mathcal{H}}^1$ agree with arc length on smooth curves in Euclidean space. Positive normalization constants never change Hausdorff dimension. A definition must specify this constant when numerical values of Hausdorff measure matter.

For a line segment of length $L$, splitting it into $n$ equal pieces gives the cover costIt tends to zero for $s>1$, stays at $L$ for $s=1$, and grows for $s<1$. The matching lower bound comes from the fact that any interval cover must have total length at least $L$. The exponent one is therefore the transition between a measuring rule too coarse to see the segment and one that assigns it infinite size. For a Cantor set, the transition is instead at $\log 2/\log 3$, as shown in Hausdorff dimension of the Cantor set.

This makes Hausdorff measure useful when ordinary Lebesgue measure sees too little: a curve in the plane has zero planar area but can have positive arc length, and the Cantor set has zero line length but positive geometric size at its own dimension. It also separates local covering estimates from global cardinality: a countable dense set has zero Hausdorff dimension, despite meeting every open region of its closure.

An original source for the construction is Felix Hausdorff's ^[ref]. The modern covering and Hausdorff dimension framework is developed in Bishop and Peres, Fractals in Probability and Analysis, §1.2.

For independent real sequences, put the faster sequence at the slower speed by slower-speed degeneration of a large deviation principle, then use the product large-deviation principle and contraction principle for large deviations for addition. Unequal speeds give a translate of the slower sequence's rate. Equal speeds give the infimal convolution of the two rates. Its unique zero is the sum of their zero points, and a translated positive finite point has positive finite cost, since compactness attains each finite fiber infimum. Thus the defining nondegeneracy is preserved.

// Target: queueing-theory.bigb

The infimum of nonconstant closed geodesic lengths in curvature $-1$, with value $+\infty$ if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus $g\ge 2$, a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.

An infimal convolution is the cheapest way to split a prescribed total between two costs. Suppose two activities contribute $u$ and $v$ to a required total $x$, costing $f(u)$ and $g(v)$. Their combined cost is optimized over all splits $u+v=x$, rather than charging both activities for the entire total.

For proper extended-real functions $f,g:X\to \mathbb{R}\cup \{+\infty \}$ on a real vector space $X$, defineThe definition does not require convexity. An infinite cost forbids that contribution, as with an indicator functional of a constraint set. If no split has two finite costs, set $h(x)=+\infty$; if the costs of admissible splits are unbounded below, $h(x)=-\infty$. Thus proper inputs need not produce a proper extended-real function. The notation $E\square F$ denotes the same operation. See Parikh and Boyd, Proximal Algorithms, §3.1 for the standard convex analysis definition and its connection to the Moreau envelope.

An infimum is a best lower bound, whereas a minimum is a value realized by some split. Writing $\min$ in place of $\inf$ requires an attaining pair $(u_{\ast },v_{\ast })$. For convex functions, mixing two nearly optimal splits and applying convexity to each cost proves that their infimal convolution is convex whenever it is proper; one must use nearly optimal splits because exact minimizers need not exist. Its convex conjugate is $f^{\ast }+g^{\ast }$ whenever the extended real number arithmetic is well defined; see conjugate of an infimal convolution for the precise identity. This turns the optimization over splits into addition in the conjugate variables.

Two quadratic costs: share the load according to its price. Let $X=\mathbb{R}$ and let the convex quadratic functions beFor fixed $x$, eliminate $v=x-u$. The derivative of the total cost is $au-b(x-u)$, so equal marginal costs giveThe cheaper activity receives more of a positive total: when $a<b$ and $x>0$, $u_{\ast }>v_{\ast }$. Completing the square gives an exact certificate for every competing split:Consequently the infimum is a unique minimum, andThe effective quadratic coefficient is $c=ab/(a+b)$, with $1/c=1/a+1/b$. It is smaller than either $a$ or $b$: allowing both activities reduces the cost compared with assigning everything to either one. If $a=b$, each receives half the total and $c=a/2$. This is a simple quadratic optimization problem whose answer preserves the form of the original costs.

Smoothing a corner by splitting its cost. Take the absolute value function $f(u)=|u|$ and the convex quadratic function $g(v)=v^2/(2t)$, with $t>0$. This infimal convolution is the Moreau envelope of the absolute value function:Here $\mathrm{sgn}$ is the sign function, with $\mathrm{sgn}(0)=0$. The optimal $u_{\ast }$ is soft thresholding. In the central region, the quadratic function carries the whole total; outside it, $v_{\ast }$ stays at $t\mathrm{sgn}(x)$ and the absolute value function carries the excess. At $t=1$ the result is the Huber loss in the normalization $x^2/2$ near zero and $|x|-1/2$ outside. Its derivative is $x$ for $|x|\le 1$ and $\mathrm{sgn}(x)$ outside, so the corner becomes a continuous change of slope. Parikh and Boyd's ^[ref] uses twice this normalization for the Huber loss; the explicit formula here fixes the convention. This is also the one-dimensional case of the existing Moreau envelope of the Euclidean norm.

For a nonempty set $A\subseteq \mathbb{R}$, a real number $\ell$ is its infimum precisely when it is a lower bound and every other lower bound is at most $\ell$. It is the greatest lower bound, not necessarily the minimum.

For $C=\{1/n:n\in {\mathbb{N}}_{>0}\}$, zero is a lower bound. Every positive candidate $\epsilon$ fails: the Archimedean property gives an integer $n>1/\epsilon$, hence $1/n<\epsilon$. Thus $\inf C=0$, although zero is never a member of $C$. This is the lower-edge counterpart of approximating an unattained supremum.

For a subset $A$ of a partially ordered set, its infimum is its greatest lower bound and its supremum is its least upper bound. An upper bound only has to lie above every member; the supremum is the smallest such bound. A lower bound lies below every member; the infimum is the largest such bound. These definitions identify the exact edges of a set even when neither edge belongs to it.

For example, $(0,1)$ and $[0,1]$ both have infimum zero and supremum one. Only $[0,1]$ has a minimum and a maximum: those must be actual members of the set. The set $C=\{1/n:n\in {\mathbb{N}}_{>0}\}$ has infimum zero without a minimum, but its supremum one is also its maximum. Such examples distinguish a bound from an attained extremal value. ^[ref].

For a nonempty subset of the real numbers, completeness of the real numbers guarantees a real supremum when it is bounded above, and a real infimum when it is bounded below. The bound and the ambient partially ordered set both matter: $\{q\in \mathbb{Q}:q\ge 0,q^2<2\}$ has supremum $\sqrt{2}$ in the real numbers, but no supremum in the rational numbers. The detailed explanation belongs to completeness of the real numbers.

Why are unattained bounds useful? They specify a precise limiting target. In mathematical optimization, $f(x)=x$ on $(0,1)$ has infimum zero but no global minimizer. The inputs $x_n=1/n$ for $n\ge 2$ give a minimizing sequence: their objective values approach zero without ever reaching it. Bounds therefore describe what approximation can achieve even when no feasible input is best. The approximation property of an infimum and approximation property of a supremum express this idea quantitatively.

For a nonnegative measurable function $f:X\to [0,\infty ]$ on a measure space, defineThe supremum may be infinite. A simple function assigns finitely many heights to measurable sets, so its integral is height times measure, summed over those pieces. This definition groups points by their values rather than requiring a partition into short intervals. See MIT lecture 11, Definition 108.

A real-valued function is Lebesgue integrable when ${\int }_X|f|d\mu <\infty$. Its integral is ${\int }_X{f}_{+}d\mu -{\int }_X{f}_{-}d\mu$, where $f_{+}=\max (f,0)$ is its positive part and $f_{-}=\max (-f,0)$. This finiteness requirement avoids subtracting two infinities. For functions taking values in the complex numbers, integrate the real part and imaginary part separately.

Consider the Dirichlet function $f=1_{\mathbb{Q}\cap [0,1]}$ on $[0,1]$. It is a simple function, andIts Riemann integral does not exist. For every partition of an interval $0=x_0<\cdots <x_n=1$, each nondegenerate $[x_{i-1},x_i]$ contains both rational numbers and irrational numbers, by their respective density of the rational numbers and density of the irrational numbers. Thus its infimum there is $0$ and its supremum is $1$. The lower Darboux sum is ${\sum }_{i}0(x_i-x_{i-1})=0$, whereas the upper Darboux sum is ${\sum }_{i}1(x_i-x_{i-1})=1$. Refinement never closes this gap. A finite scatterplot cannot represent this behavior: the assertion concerns every subinterval and every partition of an interval.

Another view builds the Lebesgue integral from horizontal layers. For any nonnegative measurable function, the layer cake representation statesBoth sides may be infinite. The right side integrates the measure of a superlevel set against Lebesgue measure in the height variable $t$. See Gautam Iyer's measure theory notes, Theorem 9.5.

For $f(x)=x^2$ on $[0,1]$ with Lebesgue measure, the superlevel set at $0\le t<1$ is $(\sqrt{t},1]$, of Lebesgue measure $1-\sqrt{t}$; for $t\ge 1$ it is empty. ConsequentlyAt height $t=1/4$, the layer extends from $x=1/2$ to $x=1$, a width of $1/2$. The picture shows why the same area can be accumulated vertically or horizontally. In a probability space, the identical formula expresses the expected value of a nonnegative random variable through its exceedance probabilities.

A proof outline avoids extra assumptions on the measure space. For a nonnegative simple function $s={\sum }_j{a}_j{1}_{A_j}$ with disjoint measurable sets $A_j$, the superlevel set has measure ${\sum }_{a_j>t}\mu (A_j)$. Integrating this finite sum gives ${\sum }_j{a}_j\mu (A_j)=\int sd\mu$. Next use approximation by nonnegative simple functions $s_n\uparrow f$. For every $t$, the superlevel sets $\{s_n>t\}$ increase to $\{f>t\}$. Continuity from below of a measure and the monotone convergence theorem on the height variable give the formula for $f$.

The limit inferior and limit superior retain the lower and upper long-run behaviour of a real sequence, even when it has no ordinary limit of a sequence. For each tail of a sequence beginning at $N$, defineDiscarding more initial terms raises or preserves the infimum and lowers or preserves the supremum. Hence $(l_N)$ is nondecreasing and $(u_N)$ is nonincreasing. Their limits exist in the extended real number line:For a bounded sequence, both values are finite and $\mathrm{liminf}{a}_n\le \mathrm{limsup}{a}_n$. These bounds use the entire infinite tail of a sequence, rather than the last few observed terms. ^[ref].

Take the original example $a_n=(-1{)}^n(1+1/n)$, beginning $-2,3/2,-4/3,5/4,\dots$. The subsequence at even integers decreases toward one, while the subsequence at odd integers increases toward minus one. If $e_N$ is the first even integer at least $N$ and $o_N$ is the first odd integer at least $N$, thenThus $\mathrm{liminf}{a}_n=-1$ and $\mathrm{limsup}{a}_n=1$. The finite initial extreme $a_1=-2$ disappears from later lower bounds; neither long-run value is attained by any term of this example.

For a bounded sequence, the limit inferior is the smallest subsequential limit and the limit superior is the largest subsequential limit; both are reached by suitable convergent subsequences. Every other subsequential limit lies between them. In the example there are only two subsequential limits, $-1$ and $1$: a convergent subsequence cannot retain infinitely many terms of both parities, because those two branches stay separated. The whole interval between the two bounds need not consist of subsequential limits. ^[ref].

A finite ordinary limit of a sequence exists exactly when the limit inferior and limit superior agree at the same finite value $L$. The bounds $l_N\le a_N\le u_N$ and the squeeze theorem prove the sufficient direction. For a bounded sequence, its tail diameter of a sequence is $D_N=u_N-l_N$, so the limiting gap is zero exactly when the Cauchy sequence condition holds. In the example the gap approaches two, not zero. The same framework allows infinite bounds: $a_n=n$ has both values $+\infty$, whereas $a_n=(-1{)}^{n}n$ has limit inferior $-\infty$ and limit superior $+\infty$. Equality at infinity describes escape rather than convergence to a real value. ^[ref].

For a stable input with path action $I(a)={\int }_{-\infty }^{0}g({\dot{a}}_s)ds$, a workload x requires some interval of length t with average input near $C+x/t$. Jensen inequality bounds its cost below by $tg(C+x/t)$; taking the supremum workload and then the infimum over t yields the displayed lower bound. Constant-rate burst paths give the matching upper bound. At zero the typical-rate path has zero cost; negative workloads have infinite rate. If no finite-cost rate exceeds C, every positive workload has infinite rate.

For a bounded function $f$ and a partition of an interval $P$, the lower Darboux sum multiplies the infimum of $f$ on each subinterval by its length and adds the results.

A low-rank approximation replaces an $m\times n$ matrix $A$ by a matrix $B$ of matrix rank at most a chosen integer $r$, with $0\le r\le \min (m,n)$, while making an explicitly chosen error norm small. The point is to describe many entries using a few shared patterns: an image compression may retain broad brightness patterns while discarding finer ones. The approximation is useful when the saved storage or computation justifies the lost detail. A small matrix rank alone says nothing about the error.

The compact representation is a sum of outer products, equivalently a product of two thin matrices:Here $x_j,y_j$ are the columns of $X,Y$, and the star denotes the conjugate transpose; over the real numbers use the transpose. The rank bound for a matrix product gives $\mathrm{rank}B\le r$. Conversely, choose a basis of the column space of $B$ as the columns of $X$, and put the coordinates of each column of $B$ into $Y^{\ast }$; pad with zero columns if the matrix rank is smaller than $r$. Each nonzero outer product is a rank-one matrix: it repeats a single column pattern at different strengths across the columns.

Keeping the two factors stores $r(m+n)$ scalars instead of $mn$. This saves space only when $r(m+n)<mn$. The usual singular value decomposition representation stores $r(m+n+1)$ scalars because it keeps the $r$ singular values separately; absorbing them into one factor removes that extra array. These are representation sizes at equal scalar precision, before file overhead. See Uschmajew and Vandereycken, §9.2: thin factors and their storage, and Illinois CS 357, Low-rank approximation: storing singular vectors and values.

The factors also give $Bz=X(Y^{\ast }z)$ without forming the full matrix $B$. Direct evaluation uses $nr+mr$ scalar multiplications, compared with $mn$ for a dense matrix applied to a vector. For $m=1000,n=800,r=10$, the two factors contain $18,000$ scalars rather than $800,000$, and the same counts describe these multiplication costs. Constructing the factors is a separate cost. Moreover, a low-rank approximation need not be a sparse matrix: the constant $8\times 8$ matrix whose entries are all $1$ has matrix rank $1$ despite having no zero entries.

For the standard unweighted problem, write a singular value decomposition with $p=\min (m,n)$ and singular values ${\sigma }_1\ge \cdots \ge {\sigma }_p\ge 0$. Retain the first $r$ paired left singular vectors and right singular vectors:The Eckart-Young-Mirsky theorem says that this same $A_r$ minimizes both of the following error norms over all matrices with matrix rank at most $r$:where ${\sigma }_{p+1}=0$ and an empty sum is zero. Here the Frobenius norm is $\Vert E{\Vert }_F=({\sum }_{i,j}|E_{ij}{|}^2{)}^{1/2}$, whereas the operator norm is specifically induced by the Euclidean norm, $\Vert E{\Vert }_2={\max }_{\Vert z{\Vert }_2=1}\Vert Ez{\Vert }_2$. The first aggregates errors in all entries; the second bounds the worst output error for a unit vector input. These formulas apply to real and complex matrices. ^[ref]

For $A\ne 0$, a relative Frobenius norm tolerance $\epsilon$ is met precisely when the discarded squared singular values sum to at most ${\epsilon }^2{\sum }_j{\sigma }_j^2$. Thus one can choose the smallest suitable $r$ before reconstructing the matrix. Many individually small discarded singular values can still have a large combined error. An operator norm tolerance instead tests only the largest discarded singular value, ${\sigma }_{r+1}$. The two questions can therefore select different ranks.

The following stipulated $8\times 8$ matrix makes the tradeoff visible. Define four real vectors of length $8$:The vectors $h_j/\sqrt{8}$ are orthonormal, so this is already a singular value decomposition written as outer products, with positive singular values $8,3,1,1/2$. All entries lie between $7/16$ and $25/16$ and can be displayed as brightnesses. The first component is uniform brightness, the second distinguishes broad blocks, and the last two supply finer alternating patterns. This is a designed arithmetic example, not measured image data.

Here $A_1=h_0{h}_0^T$ has every entry equal to $1$, while $A_2=h_0{h}_0^T+3h_1{h}_1^T/8$ has entries $11/8$ within the two diagonal blocks and $5/8$ in the other two blocks. For instance, $A_{11}=25/16$ becomes $(A_2{)}_{11}=11/8$: its error is $3/16$. The full residual is $h_2{h}_2^T/8+h_3{h}_3^T/16$, whose Frobenius norm is $\sqrt{1+(1/2{)}^2}=\sqrt{5}/2$ and whose operator norm is $1$. Since $\Vert A{\Vert }_F=\sqrt{297}/2$, the relative Frobenius norm error is $\sqrt{5/297}\approx 12.97\%$. Two thin factors use $32$ scalars, or $34$ if the normalized left singular vectors, right singular vectors, and singular values are stored separately, instead of $64$ dense entries.

Retaining three components reduces the Frobenius norm error to $1/2$ while using $48$ scalars in two factors. Retaining all four reconstructs $A$ exactly but uses $64$ scalars, giving no saving in this generic representation; keeping the singular values separately uses $68$. Our particular sign patterns could be encoded more economically, but that would exploit an additional structure beyond the general low-rank approximation. The striking feature is that numerical error and storage can both be read from the decomposition, rather than judged only by how smooth the picture looks.

The constraint is deliberately matrix rank at most $r$. The truncation has $\mathrm{rank}{A}_r=\min (r,\mathrm{rank}A)$. If $\mathrm{rank}A<r$, the best approximation is $A$ itself, with zero error. Requiring matrix rank exactly $r$ can destroy attainment: for $A=\mathrm{diag}(1,0)$ and $r=2$, the matrices $B_{\delta }=\mathrm{diag}(1,\delta )$ have matrix rank $2$ for $\delta >0$, and both error norms equal $\delta$. Their error has infimum zero, but no rank-two matrix attains it. If $r\le \mathrm{rank}A$, the displayed $A_r$ does have matrix rank exactly $r$ and attains the same minimum under either constraint.

Optimality also does not always mean uniqueness. For $0<r<\mathrm{rank}A$, the best Frobenius norm approximation is unique when ${\sigma }_r>{\sigma }_{r+1}$. ^[ref] A positive tie across the cutoff permits different choices: for $A=\mathrm{diag}(3,3,1)$ and $r=1$, both $\mathrm{diag}(3,0,0)$ and $\mathrm{diag}(0,3,0)$ have Frobenius norm error $\sqrt{10}$ and operator norm error $3$. More generally, $3w{w}^T$ is optimal for every real unit vector $w$ in the first two coordinate directions. Rotating which tied direction is kept changes the approximation. Repeated singular values entirely inside the retained group do not change its sum of outer products. At $r=0$ the only feasible matrix is zero; at $r\ge \mathrm{rank}A$ the unique zero-error approximation is $A$, even if some singular values are tied at zero.

A strict gap does not guarantee uniqueness for the operator norm. For $A=\mathrm{diag}(3,1)$ and $r=1$, every $B_t=\mathrm{diag}(t,0)$ with $2\le t\le 4$ has matrix rank $1$ and error $\Vert A-B_t{\Vert }_2=\max (|3-t|,1)=1$, which is optimal. Only $t=3$ minimizes the Frobenius norm, since $\Vert A-B_t{\Vert }_F=\sqrt{(3-t{)}^2+1}$. An unchanged worst direction can hide extra error in another direction under the operator norm.

Finally, these guarantees concern the stated norms and the sole matrix rank constraint. They do not establish that a discarded pattern is noise or irrelevant to a particular task: the finer patterns in our constructed example were put there intentionally. Adding entrywise weights or restrictions on the factors changes the problem, so the Eckart-Young-Mirsky theorem above does not automatically solve it. For historical context, the original 1936 matrix problem is documented in the existing Eckart-Young-Mirsky theorem section.