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Let $R,\alpha >0$ and ${\mathcal{C}}_{\alpha }=\{(x,t)\in {\mathbb{C}}^2:|x|<R,|t|<\alpha (1-|x|/R)\}$. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of ${\mathcal{C}}_{\alpha }$, including at $t=0$. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-derivative in a Cauchy-Kovalevskaya theorem argument.

In the weighted inner product $⟨f,g⟩={\int }_0^{1}xf(x)g(x)dx$, the best approximation to $x^2$ in the span of $1,x$ is the displayed polynomial. The normal equations are $1/4-A/2-B/3=0$ and $1/5-A/3-B/4=0$. Their solution gives $A=-3/10$, $B=6/5$. The residual is orthogonal to that span, so the Pythagorean theorem proves minimality and uniqueness. Its squared norm is $1/600$. A weight vanishing at a single endpoint still gives a positive-definite inner product on continuous functions: a nonzero continuous function remains nonzero on an interval where the weight is positive.

For the weight $w(x)=x$ on $[0,1]$, the normal equations for projecting $\sqrt{1-x^2}$ onto $\mathrm{span}\{1,x\}$ have Gram matrix $\left(\begin{array}{cc}1/2& 1/3\\ 1/3& 1/4\end{array}\right)$ and right side $(1/3,\pi /16{)}^T$. Its determinant is $1/72$, and inversion gives the displayed polynomial. orthogonality of the residual to both basis functions proves best approximation in the weighted norm.

On $(0,L)$, the differential operator $\mathcal{L}={\partial }_x^2+\beta {\partial }_x=e^{-\beta x}{\partial }_x(e^{\beta x}{\partial }_x)$ with homogeneous Neumann boundary conditions is self-adjoint in the weighted inner product with weight $e^{\beta x}$. Its eigenfunctions and nonnegative decay rates areTheir squared norms are $N_0=(e^{\beta L}-1)/\beta$ and $N_k=L[1+{\beta }^2/(4q_k^2)]/2$. At $\beta =0$, interpret the continuous limit as $N_0=L$. The Sturm-Liouville eigenfunction expansion givesThis kernel acts against the measure $e^{\beta y}dy$, not Lebesgue measure alone. It is symmetric in $x,y$; the transition density against Lebesgue measure is $e^{\beta y}{K}_{\beta }(x,y,t)$.

A positive measurable weight $w$ defines the displayed norm on the L2 space for the measure $w(x)dx$, with functions identified when equal almost everywhere. This is the norm induced by the weighted inner product. Changing $w$ changes the contribution of each region to approximation error. If $p_0,\dots ,p_N$ are nonzero mutually orthogonal real functions, completing squares shows that the unique best approximation to real $f$ in their linear span has coefficients $a_j=⟨f,p_j{⟩}_w/\Vert p_j{\Vert }_w^2$.

For a permutation $\pi$ of an orthonormal basis, the weighted image columns are mutually orthogonal and have squared norms $|w_j{|}^2$. Hence $U^{†}U=\mathrm{diag}(|w_1{|}^2,\dots ,|w_d{|}^2)$ and the displayed criterion follows. All nonzero weights suffice for invertibility, but unitarity requires unit-modulus weights. This describes basis permutations decorated by phases without introducing unwanted changes of amplitude.

For positive weights $w_j$, the sequence space with norm $\Vert \alpha \Vert ={\sum }_j{w}_j|{\alpha }_j|$ is isometrically isomorphic to ${\ell }_1$: multiplication by $w_j$ is a linear bijection onto the absolutely summable sequence space, with inverse multiplication by $w_j^{-1}$. Thus it is a separable Banach space. Density can also be seen directly by truncating a weighted summable sequence and then approximating its finitely many coordinates by rational numbers.

The Wiener algebra consists of periodic functions with absolutely summable Fourier coefficients, with norm ${\sum }_k|a_k|$. Multiplication corresponds to discrete convolution of coefficients, so the norm is submultiplicative. Consequently $e^f={\sum }_{n\ge 0}{f}^n/n!$ also belongs to the algebra when $f$ does. Every such Fourier series is uniformly convergent and defines a continuous function.