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The cellwise error of the Haar projection estimator in Gaussian white noise is $\sqrt{2^J/n}$ times a standard normal variable. Its supremum norm is therefore exactly that factor times the Gaussian maximum over $2^J$ cells. The Gaussian maximum bound without independence gives expected error at most $\sqrt{2^J(2J+2)\log 2/n}$. This concerns the stochastic error about the projection; approximation bias must be added when estimating the full drift.

On a nondegenerate interval, point evaluation of the derivative is unbounded for the supremum norm on smooth functions. The functions $f_N(x)=\sin (N(x-x_0))$ have supremum norm at most one but $f_N^{\prime }(x_0)=N$. Adding the supremum norm of the derivative makes this evaluation a bounded linear functional.

A uniform algebra on a compact Hausdorff space $K$ is a closed complex subalgebra of $C(K)$, with the supremum norm, containing the constants and separating points. Evaluation embeds $K$ continuously and injectively in the character space of an algebra. The embedding is a homeomorphism onto a closed subset because its domain is compact and its target is Hausdorff. Extra characters need not be point evaluations on the specified $K$.

Uniform approximation minimizes the supremum norm of the difference between a target function and an approximant.

The error is measured uniformly over the domain using the supremum norm. A sequence of approximants whose uniform error tends to zero converges by uniform convergence; the optimization problem and the convergence notion are distinct.

For fixed $r$ and $f\in C^r[0,1]$, uniform convergence of normalized finite differences to $f^{(r)}$, uniform continuity of $f^{(r)}$, and $(n{)}_r/n^r\to 1$ imply $g_{n,r}\to f^{(r)}$ uniformly. Positivity and partition of unity make Bernstein polynomials contractions in the supremum norm. Consequently $\Vert B_{n-r}{g}_{n,r}-f^{(r)}{\Vert }_{\infty }\le \Vert g_{n,r}-f^{(r)}{\Vert }_{\infty }+\Vert B_{n-r}{f}^{(r)}-f^{(r)}{\Vert }_{\infty }\to 0$. The differentiability assumption is essential; continuity alone does not define the target derivative.

If sequences of functions $f_n\to f$ and $g_n\to g$ have uniform convergence on the same domain and each term is a bounded function, their products converge uniformly to $fg$. The two families are uniformly bounded families of functions: one late term and the Uniform Cauchy criterion bound the entire tail, while finitely many early terms have a finite common bound. Write $f_n{g}_n-fg=f_n(g_n-g)+g(f_n-f)$ and take the supremum norm to obtain the displayed estimate. The boundedness hypothesis cannot be dropped: $f_n(x)=x$ and $g_n(x)=1/n$ converge uniformly on $\mathbb{R}$, whereas $x/n$ fails to converge uniformly to zero.

For each positive integer $n$, put $f_n(t)=\sin (2\pi nt)/n$ on $[0,1]$. Its supremum norm is exactly $1/n$, so this sequence of functions converges to zero by uniform convergence. Nevertheless,Each of its $n$ complete sine oscillations contributes $4/n$: rise from zero to $1/n$, fall through zero to $-1/n$, then return to zero. Taking a partition of an interval through all extrema attains $4$. Every further partition refinement merely subdivides monotone function segments, so the same reasoning as in partition example of total variation proves the upper bound too.

For a continuously differentiable function on a closed bounded interval, total variation also satisfiesApplied here, the derivative is $2\pi \cos (2\pi nt)$, and integrating its absolute value over $n$ periods gives $4$. The differentiability hypothesis matters: the partition definition applies much more widely than this derivative formula.^[ref] Small pointwise size and small accumulated movement are different approximation goals.

A uniformly continuous real function on a closed bounded interval can be approximated uniformly by finite step functions. Given $\epsilon >0$, choose a mesh smaller than the uniform-continuity distance and set the approximant equal to a sampled value on each interval. Then the supremum norm of the approximation error is at most $\epsilon$. If the function has a Lipschitz bound $L$, a mesh of maximum length $h$ gives the explicit error bound $\Vert f-g{\Vert }_{\infty }\le Lh$.

Evaluation at zero is a bounded homomorphism on $C[0,1]$. Its kernel $I$ is a closed self-adjoint ideal and hence a C-star subalgebra. It has no identity: applying a proposed identity to $f(t)=t$ would force that identity to equal one at every positive point, contradicting continuity and its value zero at zero. The map $(f,\lambda )\mapsto f+\lambda 1$ from the unitization of a C-star algebra is a bijective star homomorphism, with inverse $h\mapsto (h-h(0),h(0))$. Pulling back the supremum norm verifies the C-star identity, so this is the prescribed unitization norm. Thus adjoining the missing constant direction recovers all continuous functions on the interval.

Let $C$ be closed in a normal topological space and $g:C\to [-M,M]$ continuous. Separate its closed level sets $g\le -M/3$ and $g\ge M/3$ by Urysohn's lemma to obtain $h:X\to [-M/3,M/3]$ taking the corresponding endpoint values. Checking the outer and middle ranges gives the displayed bounds. Iteration on residuals gives a uniformly convergent series with total supremum norm at most $M$. The uniform limit theorem gives the bounded Tietze extension theorem.

Weak convergence to zero gives pointwise convergence by the continuous evaluation functionals and uniform boundedness in the supremum norm by the weakly bounded set criterion. The dominated convergence theorem for Lebesgue measure then proves the displayed implication. The finite measure of the interval and uniform norm bound are the relevant hypotheses.

On $[0,1]$, the polynomials $p_n(x)=x^n/n$ converge in supremum norm to the zero function, since $\Vert p_n{\Vert }_{\infty }=1/n$. But $p_n^{\prime }(1)=1$ for every $n$, whereas the zero function has left derivative zero at that endpoint. Thus uniform convergence of function values does not imply uniform convergence of derivatives. This does not contradict uniform convergence of Bernstein polynomial derivatives, which has a particular approximation construction and additional differentiability hypotheses.

Every continuous real-valued function on a compact interval can be approximated by polynomials with arbitrarily small error everywhere on the interval. More precisely, if $a<b$ and $f:[a,b]\to \mathbb{R}$ is continuous, then for every $\epsilon >0$ there is a polynomial $p$ withThe error is measured in the supremum norm, so one approximation works at every point. The degenerate interval $a=b$ needs only a constant function. A change of variable $t=(x-a)/(b-a)$ reduces the nondegenerate case to $[0,1]$.^[ref]

This is useful because a polynomial is described by finitely many coefficients and can be evaluated with addition and multiplication. Its surprise is that the target need not be differentiable: the sharp corner of an absolute value function can be approximated by smooth polynomials. The Bernstein approximation of a corner below makes this visible. Approximating values, however, does not by itself approximate derivatives; Weierstrass approximation does not preserve derivatives gives a counterexample.

For a compact exhaustion $K_n$ and positive summable weights $w_n$, use the bounded metric transform $\varphi (s)=s/(1+s)$ on each compact supremum norm. The resulting series defines a metric on continuous functions and realizes local uniform convergence on compact subsets. The triangle inequality is termwise. For convergence, control finitely many compact seminorms and then the uniformly bounded weight tail; conversely a vanishing metric controls each fixed seminorm.