Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus

Throughout we work in the setting of Real Hilbert Spaces: Standing Notation and Background, whose notation is in force for every real inner product space and every real Hilbert space named in a result adopting this setting. This setting adds the standing notation for series, products, orthonormal bases and differential calculus on those spaces, and introduces no concepts of its own.

1. (Series)¶ Series of real numbers and series in a real inner product space, with their partial sums, their sums and absolute convergence, are as defined there; Elementary Properties of Series of Real Numbers, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series, Elementary Properties of Series in a Real Inner Product Space and Products and Sums of Weighted Square-Summable Sequences of Real Numbers are in force.

2. (Products)¶ For real inner product spaces $E_{1}$ and $E_{2}$, $E_{1}\times E_{2}$ denotes their product, with the coordinate maps $\pi_{1},\pi_{2}$ and coordinate injections $j_{1},j_{2}$ fixed there; Properties of the Product of Two Real Inner Product Spaces is in force, so that in particular the product is a real inner product space carrying the norm and distance recorded in claim 1 of that lemma.

3. (Orthonormal bases)¶ An orthonormal basis of a real Hilbert space is as defined there, and Orthonormal Expansions in a Real Hilbert Space, The Subspaces Spanned by an Orthonormal Sequence and Exhausting Sequences and A Real Hilbert Space with an Orthonormal Basis is Separable are in force.

4. (Differential calculus)¶ For a real inner product space $E$ named in a result adopting this setting, an open subset $U$ of $E$ and a function $u:U\to\mathbb{R}$: that $u$ is differentiable at a point of $U$ with gradient $Du(x)\in E$, that $u$ is differentiable on $U$ with gradient map $Du:U\to E$, that $u$ has a second derivative at a point, its Hessian $D^{2}u(x)\in\mathrm{Sym}(E)$ there, its Hessian map $D^{2}u:U\to\mathrm{Sym}(E)$, and the classes $C^{1}(U)$ and $C^{2}(U)$ are as defined there. The results Vanishing of Uniformly Small Linear, Quadratic and Bilinear Terms in a Real Inner Product Space, Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space, Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ and Segment Derivatives, the Second-Order Taylor Expansion, and the Second-Order Condition at a Local Extremum are in force, and, when $E$ is a Hilbert space, so is The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space.