Past exam of the functional analysis course of the University of Oxford / 2011 / 3

A biorthogonal system is a family $(x_i,f_i{)}_{i\in I}$ with $x_i\in X$, $f_i\in X^{\ast }$, satisfyingThus each continuous coordinate functional takes value one on its own vector and zero on all others. The vectors need not be orthogonal in any inner product, and totality of their linear span is an additional condition.

Fix $i$ and let $M_i=\overline{\mathrm{span}}\{x_j:j\ne i\}$. The hypothesis gives $M_i\ne X$. Moreover $x_i\notin M_i$, since otherwise the closed span of all the vectors would be contained in $M_i$. Hence $d_i=\mathrm{dist}(x_i,M_i)>0$.

On $M_i+\mathbb{R}{x}_i$, define ${\varphi }_i(m+t{x}_i)=t$. It is well-defined because $x_i\notin M_i$. For $t\ne 0$,so $|{\varphi }_i(m+t{x}_i)|\le \Vert m+t{x}_i\Vert /d_i$; the inequality also holds for $t=0$. The Hahn-Banach theorem extends ${\varphi }_i$ without increasing its norm to $x_i^{\ast }\in X^{\ast }$. This gives $x_i^{\ast }(x_j)={\delta }_{ij}$.

If two continuous functionals have these values, their difference vanishes on the linear span of every $x_j$, which is dense in $X$. Continuity then makes the difference zero everywhere. Thus each functional is unique, proving the required unique biorthogonal system. In fact its norm is $\Vert x_i^{\ast }\Vert =1/d_i$, by taking vectors in $M_i$ approaching the distance infimum. This is the coordinate-functional construction for a minimal total family in a normed space.

Define the finite-rank operatorIt is linear and bounded, with $\Vert P_n\Vert \le {\sum }_{i=1}^n\Vert x_i^{\ast }\Vert \Vert x_i\Vert$. Biorthogonality gives exactly the two required values on $x_j$. It also givesThus $P_n$ is a bounded projection onto $\mathrm{span}\{x_1,\dots ,x_n\}$, with kernel ${\bigcap }_{i=1}^n\ker x_i^{\ast }$. Its action on the dense span determines it uniquely among bounded linear operators. These are the partial-sum projections; no uniform bound in $n$ has yet been assumed.

Assume first that $\Vert P_n\Vert \le M$ for all $n$. Given $x\in X$ and $ϵ>0$, choose a finite linear combination $y$ of the $x_i$ with $\Vert x-y\Vert <ϵ/(M+1)$. For all sufficiently large $n$, $P_{n}y=y$. ThereforeThus $P_{n}x\to x$, and the formula in part (iii) yields the norm-convergent expansionThis proves the stated existence of coefficients.

Conversely, suppose every $x$ has some norm-convergent expansion $x={\sum }_i{\alpha }_i{x}_i$. Applying the continuous functional $x_j^{\ast }$ to its partial sums and passing to the limit gives ${\alpha }_j=x_j^{\ast }(x)$. Hence those partial sums are precisely $P_{n}x$, so $P_{n}x\to x$ for every $x$. In particular ${\sup }_n\Vert P_{n}x\Vert <\infty$ for each fixed $x$. The Uniform boundedness principle, using completeness of $X$, now givesThis proves the reverse implication. The coefficients are automatically unique, so the equivalence is the uniform partial-sum criterion for a Schauder basis for the given minimal total sequence. Its projection bound is the basis constant.