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)(i) The notation and background of The Real Numbers: Standing Notation and Background are in force. In particular $\mathbb{R}$ is the ordered field of real numbers with its order, its absolute value $|s|$, its quotients $\tfrac{s}{t}$ by a nonzero $t$, its natural powers $s^{n}$ and the metric $d_{\mathbb{R}}(s,t)=|s-t|$; $\mathbb{N}$ is the set of natural numbers, $[n]$ the initial segment of $n\in\mathbb{N}$ and $X^{n}$ the set of $n$-tuples in a set $X$; finite sums of real numbers, sequences and their subsequences, convergence and the Cauchy condition for sequences of real numbers, and suprema and infima of sets of real numbers and of real-valued functions are as fixed there. Finite sums of vectors are the finite sums in a vector space, whose properties are those of Properties of Finite Sums of Vectors. The maximum $\max\{s,t\}$ and the minimum $\min\{s,t\}$ of two real numbers are as defined there, with Elementary Properties of the Maximum of Two Elements and Elementary Properties of the Minimum of Two Elements in force.

2. (The space)(ii) $H$ 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 (iii), 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)(iii) 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 Spaces (ii) and Real Hilbert Spaces (iii). Continuity of a map defined on a subset $A$ of one of the metric spaces named in this setting, with values in one of them or in $(\mathbb{R},d_{\mathbb{R}})$, is continuity at a point relative to $A$, and on $A$, taken with the ambient metric space in which $A$ lies, and is equivalent to sequential continuity by Continuity Between Metric Spaces is Equivalent to Sequential Continuity; Lipschitz maps are as defined there. A subset $A$ of one of the metric spaces named in this setting is itself regarded as a metric space under the restriction of that metric to $A$, which is a metric by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology; it is written $(A,d)$ with the name of the ambient metric, and a subset of $A$ is called closed in $(A,d)$ when it is closed in this metric space. Open balls $B_{d}(x,r)$ and closed balls $\bar{B}_{d}(x,r)$ in these metric spaces are as defined there, with Open Ball in a Metric Space is Open and Elementary Properties of the Closed Ball in a Metric Space in force; the open subsets of such a metric space form a topology by The Open Subsets of a Metric Space Form a Topology, so that in particular each of these metric spaces is an open subset of itself. Uniform continuity of a map on a subset $A$ is as defined there, and a modulus of continuity is a function on the set of nonnegative real numbers as defined there, with The Nondecreasing Envelope of a Truncated Modulus of Continuity and Linear Moduli of Continuity in force; as fixed by that definition, a modulus of continuity is not required to be nondecreasing. For a function $u$ with domain $A$ and a subset $B$ of $A$, $u|_{B}$ denotes the restriction of $u$ to $B$.

4. (Bounded linear maps)(iv) 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)(v) $\mathrm{Sym}(E)$ 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 $b|_{E_{0}}$ to a linear subspace $E_{0}$ 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 (iv). The zero form $0_{\mathrm{Sym}}$, whose value at every pair of elements of $E$ is $0$, is the zero vector of the vector space $\mathrm{Sym}(E)$ of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i).

6. (Orthogonality and projections)(vi) 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 (i), which is linear by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space (ii) and satisfies $|P_{N}x|\le|x|$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space (iii), so that $P_{N}\in\mathcal{L}(H)$.

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

8. (Separability and exhausting sequences)(viii) 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. (Semicontinuity)(ix) For a subset $A$ of one of the metric spaces named in this setting, a function $u:A\to\mathbb{R}$ is upper semicontinuous or lower semicontinuous at a point of $A$ relative to $A$, and on $A$, as defined there, and Sums and Nonnegative Multiples of Semicontinuous Functions, Semicontinuity Under Negation and Characterization of Continuity, Semicontinuity via Sublevel and Superlevel Sets and Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space are in force. Superlevel sets, and the property of having closed superlevel sets in an ambient metric space, are as defined there, with Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits in force.

10. (Sequentially strict extrema)(x) Sequentially strict maxima and sequentially strict minima of a real-valued function on a subset of a metric space are as defined there, with Elementary Properties of Sequentially Strict Extrema in force.

11. (Background)(xi) 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.

12. (Local extrema)(xii) For a subset $A$ of one of the metric spaces named in this setting and a function $u:A\to\mathbb{R}$, a local maximum or a local minimum of $u$ at a point of $A$ relative to $A$ is as defined there, with the ambient metric space in which $A$ lies.

13. (Semicontinuous envelopes)(xiii) For a nonempty subset $S$ of one of the metric spaces named in this setting and a function $u:S\to\mathbb{R}$, that $u$ is bounded above near each point of $S$, or bounded below near each point of $S$, is as defined there, with the ambient metric space in which $S$ lies; for such $u$ the upper semicontinuous envelope $u^{*}:S\to\mathbb{R}$ and the lower semicontinuous envelope $u_{*}:S\to\mathbb{R}$ are as defined there, and Properties of the Upper Semicontinuous Envelope and Properties of the Lower Semicontinuous Envelope, by Duality are in force.