Real Hilbert Spaces: Standing Notation and Background

This setting fixes the standing notation used by results on real Hilbert spaces and by the theory of viscosity solutions built on them. It introduces no new concepts.

1. (Numbers and sequences)¶ $\mathbb{R}$ is the ordered field of real numbers, with the notation of that item, so that natural numbers are read in $\mathbb{R}$ through the canonical map, whose properties are those of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; $|s|$ is the absolute value of $s\in\mathbb{R}$ and $d_{\mathbb{R}}(s,t)=|s-t|$ the metric on $\mathbb{R}$ of The Absolute Value Metric on the Real Line. $\mathbb{N}$ is the set of natural numbers, $[n]$ the initial segment of $n\in\mathbb{N}$, $X^{n}$ the set of $n$-tuples in a set $X$, and finite sums of real numbers and of vectors are the finite sums in a field and the finite sums in a vector space. A sequence is indexed by $\mathbb{N}$, its subsequences are as defined there, and a sequence of real numbers converges as defined there.

2. (The space)¶ $H$ denotes a real Hilbert space, with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$, a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric, and zero vector $0_{H}$; when several such spaces are in play the ambient notation $\langle\cdot,\cdot\rangle_{H}$, $|\cdot|_{H}$, $d_{H}$ is used. Sums, differences and scalar multiples of vectors are those of the vector space $H$ over $\mathbb{R}$. The notation of clauses 2 to 5 and 7 is fixed, with the same meaning, for every real inner product space named in a result adopting this setting, and that of clause 6 for those spaces the result calls Hilbert spaces; completeness is asserted only of the latter.

3. (Topological vocabulary)¶ Open, closed, dense and bounded subsets of $H$, its closure operation, convergent and Cauchy sequences in $H$, and closed linear subspaces are as fixed in Real Hilbert Space §topology and Real Hilbert Space §closed-subspace. Continuity of a map between such spaces, or into $(\mathbb{R},d_{\mathbb{R}})$, is continuity between metric spaces, equivalent to sequential continuity by Continuity Between Metric Spaces is Equivalent to Sequential Continuity; Lipschitz maps are as defined there.

4. (Bounded linear maps)¶ For real inner product spaces $E$ and $F$, $\mathcal{L}(E,F)$ is the set of bounded linear maps from $E$ to $F$, $\mathcal{L}(E)=\mathcal{L}(E,E)$, $\mathrm{id}_{E}$ is the identity map, $\lVert T\rVert$ the operator norm, and bounded linear functionals are as defined there.

5. (Bounded symmetric bilinear forms)¶ $\mathrm{Sym}(E)$ is the set of bounded symmetric bilinear forms on $E$, with its norm $\lVert b\rVert$, its order $\preceq$, the identity form and the operations on forms, namely $I=I_{E}$, its multiples $cI$, and sums and scalar multiples of forms, and the restriction $b|_{W}$ to a linear subspace $W$ of $E$ carrying a stronger inner product; $d_{\mathrm{Sym}}(b_{1},b_{2})=\lVert b_{1}-b_{2}\rVert$ is a metric on $\mathrm{Sym}(E)$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §metric.

6. (Orthogonality and projections)¶ For a subset $M\subseteq H$, $M^{\perp}$ is its orthogonal complement; orthonormal tuples and sequences are as defined there. For a closed linear subspace $N$ of $H$, $P_{N}:H\to H$ is the orthogonal projection onto $N$, the nearest-point map of Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space §existence, which is linear by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §linear and satisfies $|P_{N}x|\le|x|$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §pythagoras, so that $P_{N}\in\mathcal{L}(H)$.

7. (Weak convergence)¶ $x_{m}\rightharpoonup x$ denotes weak convergence of a sequence.

8. (Separability and exhausting sequences)¶ Separability of $H$ refers to the metric space $(H,d)$ being separable. Exhausting sequences $(H_{n})_{n\in\mathbb{N}}$ for $H$ are as defined there; for such a sequence, $P_{n}=P_{H_{n}}$ and $Q_{n}:H\to H$, $Q_{n}x=x-P_{n}x$, denote its projections and tails, as in Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections.

9. (Background)¶ The following results are in force by reference for every space named in a result adopting this setting: Elementary Identities in a Real Inner Product Space, The Cauchy-Schwarz Inequality in a Real Inner Product Space, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space, Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space, Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space, Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces, The Riesz Representation Theorem for a Real Hilbert Space, Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity, Gram-Schmidt Orthonormalisation in a Real Inner Product Space, Projection onto the Span of an Orthonormal Tuple, and Coordinates on a Finite-Dimensional Subspace, Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections, Elementary Properties of Weak Convergence in a Real Inner Product Space and Bounded Sequences in a Separable Real Hilbert Space Have Weakly Convergent Subsequences.