Real Normed, Inner Product and Hilbert Spaces: Notation

(i) Base. This setting extends Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices: its conventions and notation are in force.

(ii) Vector spaces. Real vector spaces are as in Vector Space over a Field. In a real vector space $V$, the sum of $u,v\in V$ is $u+v$ and the scalar multiple of $v$ by $\lambda\in\mathbb{R}$ is $\lambda v$; $0_V$ is the zero vector, $-v$ the negative of $v$ and $u-v$ the difference. Background: Uniqueness of the Zero Vector and of Negatives and Elementary Identities in a Vector Space.

(iii) Normed spaces. Norms and real normed spaces are as in Norms and Real Normed Spaces; the norm of a real normed space $V$ is written $\|\cdot\|$, or $\|\cdot\|_V$ when several spaces are in play, and $d_V$ is its norm metric. Convergence and limits of sequences in $V$, Cauchy sequences in $V$, and open, closed and bounded subsets of $V$ refer to the metric space $(V,d_V)$, closed subsets being those closed in $(V,\mathcal{T}_{d_V})$, with the notation of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (ii) and Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iii). Background: The Norm Metric of a Real Normed Space.

(iv) Inner product spaces. Inner products, real inner product spaces and the induced norm $\|\cdot\|$ are as in Real Inner Product Spaces, the inner product being written $\langle\cdot,\cdot\rangle$, or $\langle\cdot,\cdot\rangle_V$. A real inner product space $V$ is treated as the real normed space $V$ with its induced norm, and the conventions of (iii) apply to it. Background: The Cauchy–Schwarz Inequality in a Real Inner Product Space and The Norm of a Real Inner Product Space, in particular The Norm of a Real Inner Product Space (iii).

(v) Banach and Hilbert spaces. Real Banach spaces and real Hilbert spaces are as in Real Banach Space and Real Hilbert Spaces. For $n\in\mathbb{N}$, $\mathbb{R}^{n}$ with the dot product is treated as a real Hilbert space. Background: ℝⁿ Is a Real Hilbert Space, in particular ℝⁿ Is a Real Hilbert Space (ii) and ℝⁿ Is a Real Hilbert Space (iii).