Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $d\in\mathbb{N}$, let $(H,M,\Omega)$ be a tracial W*-probability space, let $X,Y$ be $L^{2}$ $d$-tuples of $(H,M,\Omega)$, and let $t$ be real. Let $H^{d}$ be the complex Hilbert space of $d$-tuples in $H$ of Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert, which contains every $L^{2}$ $d$-tuple.

1. (Sums and real multiples)¶ The sum and the real multiple are the $d$-tuples $X+Y=(X_{1}+Y_{1},\dots,X_{d}+Y_{d})$ and $tX=(tX_{1},\dots,tX_{d})$, that is, the sum and the scalar multiple in $H^{d}$; they are $L^{2}$ $d$-tuples by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint.

2. (Pairing)¶ The pairing of $X$ and $Y$ is $\langle X,Y\rangle_{2}=\sum_{j=1}^{d}\langle X_{j},Y_{j}\rangle$, the inner product of $X$ and $Y$ in $H^{d}$; it is a real number because each $\langle X_{j},Y_{j}\rangle$ is real by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint. The $L^{2}$ norm $\lVert X\rVert_{2}$ of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples is the norm of $X$ in $H^{d}$ by Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert.