Explanation of orthonormal basis and isomorphism

Question

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What does the fancy $L^2(B)$ represent, and can someone explain this statement? Why is $\langle (\phi(x) , \phi(y) \rangle= \langle x,y \rangle$?

Answer 1

For any set $B$ and any function $f: B \to \mathbb R^{+}$ we define $\sum_{x \in B} f(x)$ as the supremum of all finite sums $\sum f(b_i)$ where $(b_i)$ is a finite subset of $B$. $\ell^{2} (B)$ is the set of all functions $f: B \to \mathbb R$ such that $\sum_{x \in B} |f(x)|^{2} <\infty$ with the norm $\|f\|=\sqrt { \sum_{x \in B} |f(x)|^{2}}$.

If $x \in H$ then $x =\sum_B \langle x, b \rangle b$ where all but countable many terms are $0$. Associate with $x$ the function $f(b) =\langle x , b \rangle$ With these definition it follows immediately that the map so defined is an isometric isomorphism from $H$ onto $\ell^{2}(B)$.