For the Banach space $c$ of convergent complex sequences with the supremum norm, every bounded complex-linear functional has the unique representationRestrict $F$ to $c_0$ and use coordinate duality of sequence spaces to obtain $(a_j)\in {\ell }^1$. Splitting $x$ into its constant limit sequence and a zero-limit remainder forces $a_{\infty }=F(1)-{\sum }_j{a}_j$. For the norm lower bound choose the first $N$ coordinates of $x$ with phases maximizing $a_j{x}_j$ and make its tail the constant phase maximizing $a_{\infty }\lim x$. The finite prefix and limit contributions attain their absolute values; the remaining contribution is bounded by the coefficient tail, which tends to zero. The opposite norm inequality is the triangle inequality. The index at infinity is an independent coordinate of the dual norm, rather than part of the sum of the finite-coordinate coefficients.

For a normed vector space $X$, a family $(x_i,f_i{)}_{i\in I}$ in $X\times X^{\ast }$ is biorthogonal when the displayed Kronecker-delta equations hold. Its vectors are linearly independent: applying $f_i$ to a finite linear dependence isolates its $i$th coefficient. No inner product is required, and the family need not span a dense subspace. If its linear span is dense, each prescribed coordinate functional is unique by continuity. For a countable total family, the finite sums $P_{n}x={\sum }_{i\le n}{f}_i(x)x_i$ are bounded projection operators, since biorthogonality gives $P_n^2=P_n$.

For the basis $v_j=e_1+\cdots +e_j$ obtained from a basis $e_1,\dots ,e_n$, the dual basis is obtained by successive differences of the original dual basis. Indeed, if $x={\sum }_j{c}_j{v}_j$, its original coordinates are $x_i={\sum }_{j\ge i}{c}_j$. Hence $c_i=x_i-x_{i+1}$ for $i<n$, and $c_n=x_n$. These are precisely the new coordinate linear functionals. This gives an elementary example of the inverse relation between change of basis and covector transformation under a change of basis.

For $p(X)=a+bX+c{X}^2$, the three point evaluation functionals at $0,1,-1$ form a basis of the dual space of real polynomials of degree at most two. The displayed recovery formulas express the coordinate functionals, hence the dual basis to $1,X,X^2$, as linear combinations of these evaluations. More generally distinct-node polynomial interpolation identifies values with coefficient data through an invertible Vandermonde matrix.

Suppose $u$ and $\log (1+x)u$ are Lebesgue integrable functions on $(0,\infty )$. For integers $n\ge 1$, $x+n\le n(1+x)$ gives the displayed L1 norm bound. It tends to zero because $\log n/n\to 0$, proving L1 convergence of the logarithmically weighted functions to zero. Alternatively, $|\log (n+x)u(x)/n|\le (1+\log (1+x))|u(x)|$ and pointwise convergence permit the dominated convergence theorem.

For $u(x)=\sin x/x^q$ with $1<q<2$, both integrability conditions hold. Near zero, $|\sin x|\le x$ and $\log (1+x)\le x$ give integrable bounds by powers. At infinity, $|\sin x|\le 1$ and $\log (1+x)\le \log 2+\log x$, with ${\int }_1^{\infty }{x}^{-q}\log xdx=(q-1{)}^{-2}$. The L1 norm consequently decays as $O(\log n/n)$ in Big O notation.

If $f$ is jointly continuous on a compact rectangle $J\times [a,b]$, then the displayed parameter integral is continuous on $J$. For $t_n\to t$, pointwise continuity gives $f(t_n,x)\to f(t,x)$. The extreme value theorem bounds $|f|$ by a constant $M$ on the rectangle, and $M$ is integrable on the finite interval $[a,b]$. The dominated convergence theorem gives $F(t_n)\to F(t)$.

The same conclusion holds on an unbounded parameter interval if $f$ is jointly continuous there: use a compact rectangle around each fixed parameter value. A global bound over the whole parameter range is unnecessary.

For real $a$, the geometric-series expansion of $1/(e^x-1)$ gives an integrable series after multiplication by $\sin (ax)$: each absolute integral is bounded by $|a|{\int }_0^{\infty }x{e}^{-nx}dx=|a|/n^2$. Termwise integration is therefore justified and gives the convergent series ${\sum }_{n\ge 1}a/(a^2+n^2)$.

Use of heat to reduce viable microorganisms. Temperature, exposure time and the material’s thermal behavior affect the result, so a hot surface does not by itself establish the treatment achieved its intended endpoint.

A random variable whose values lie in the integers with probability one. Its probability mass function is $p(j)=\mathbb{P}(X=j)$ for $j\in \mathbb{Z}$. If it is nonnegative, its probability generating function is $G(s)={\sum }_{j\ge 0}p(j)s^j$ wherever this expectation is finite.

For $0<\theta <1$, this probability distribution on positive integers has mass $\mathbb{P}(X=k)={\theta }^k/(k[-\log (1-\theta )])$. The Taylor identity ${\sum }_{k\ge 1}{\theta }^k/k=-\log (1-\theta )$ proves normalization. Its expectation is $-\theta /((1-\theta )\log (1-\theta ))$, by summing the geometric series after multiplying the mass by $k$.

For an independent sample, the maximum-likelihood estimator solves the equation making this model mean equal to the sample mean. The model mean is strictly increasing: set $t=-\log (1-\theta )$ to obtain $(e^t-1)/t$, whose derivative is positive for $t>0$. Hence a sample mean greater than one determines a unique interior estimate. Pooling several categories changes the likelihood and generally changes this estimate; the ungrouped sample mean is no longer recoverable from the pooled category counts alone.

A directed cycle in an edge-weighted graph whose total weight is negative. Repeating it makes walk costs arbitrarily small. For a particular shortest-path distance $d(s,t)$ to be affected, the cycle must be reachable from $s$ and must have a route to $t$; an unrelated component does not change that distance.

Among paths of minimum total weight, choose the fewest edges. Store a lexicographic order pair $(d,h)$ and extend it across weight $w$ as $(d+w,h+1)$. For positive edge weights, every extension strictly increases its first component, so Dijkstra's algorithm settles these pairs correctly when its priority queue and relaxation both compare the full pair. Equal total weight then selects the smaller hop count; arbitrary vertex-name tie-breaking does not express this objective.

Process a digit-array integer representation from low to high positions. The integer division supplies $q=\lfloor a_i/b\rfloor$ and $0\le r<b$. Replacing the current digit by $r$ and adding $q$ to the next position preserves the represented value. Append zero storage when a positive carry leaves the allocated array. The processed prefix remains normalized and the value stays fixed.

This also works with signed temporary digits when the represented value is nonnegative. A negative carry cannot leave the final allocated position: the normalized lower digits represent a value less than that position's next radix power, which could not offset that negative carry. Positive carries eventually stop beyond the value's finite radix expansion. Thus the usual nonnegative-digit algorithm admits this useful extension without interpreting truncated division as floor division.

When the radix is $10^k$, print the highest nonzero normalized digit without leading zeros, then every lower digit as exactly $k$ decimal characters. Internal zero blocks must be retained. With base $1000$, the low-first digits $[4,0,12]$ print as 12 followed by 000 followed by 004, giving 12000004. An all-zero representation prints a single zero, independent of its allocated length.

A deterministic finite automaton whose automaton state is a pair of automaton states of two given deterministic finite automata, updated componentwise by each input symbol. Choosing accepting pairs by a Boolean function on component acceptance realizes intersection, union, or set difference of the recognized regular languages.

In the Leibniz formula for a determinant of a block diagonal matrix, a term is zero unless its permutation maps every block's row indices onto that same block's column indices. The surviving permutations are precisely independent permutations within the blocks; their signs and products of entries factor over the blocks. Summing therefore gives the product of the individual determinants. Applied to $tI-T$, this also proves that the characteristic polynomial of a block diagonal linear operator is the product of the characteristic polynomials of its blocks.

Foundation implies that no descending membership sequence exists. Conversely, with dependent choice, failure of foundation gives a nonempty set in which every element contains another element of that set. Successively selecting such members produces a descending membership sequence. Thus the two conditions are equivalent with dependent choice; full choice is sufficient.

For a solvable group, its derived length is the least nonnegative integer $r$ at which the derived series reaches the trivial group, with $G^{(0)}=G$ and $G^{(r+1)}$ the commutator subgroup of $G^{(r)}$. A nontrivial abelian group has derived length one; a nonabelian group with nontrivial abelian commutator subgroup has derived length two.

For $t$ in a compact positive interval $[a,b]$, split the integration region at one. Near zero the smallest exponent gives the displayed bound, whose integral is $1/a^2$. Above one the largest exponent gives the bound, integrable by exponential decay. A single integrable function therefore dominates all parameter derivatives locally. Dominated differentiation under the integral sign gives ${\Gamma }^{\prime }(t)={\int }_0^{\infty }{e}^{-x}{x}^{t-1}\log xdx$. The same argument with higher powers of $|\log x|$ gives all higher derivatives.

The right derivative is the displayed one-sided limit, when it exists as a finite number. It describes differentiation at the left endpoint of a closed interval without requiring the function to be defined to the left. An ordinary derivative at an interior point requires matching left and right derivatives. For example, the absolute value function has right derivative one at zero and left derivative minus one, so it has no ordinary derivative there.