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Let $\mathcal{H}$ an arbitrary Hilbert space no necessary Separable.

Let $\{u_i:i\in I\}$ a orthonormal basis of $\mathcal{H}$ where $I$ is a uncountable set.

Let $\mathbb{K}$ a field and $l_2(I)$ the spaces the sequences $(x_i)_{i\in I}$ with $x_i\in\mathbb{K}$ so that $\sum_{i\in I}|x_i|^2$ is summable.

I need to show that $\mathcal{H}$ and $l_2(I)$ are isometrically isomorphism.

For this, let $\varphi:\mathcal{H}\to l_2(I)$ defined by $\varphi(x)=\{\langle x,u_i\rangle\}_{i\in I}$. Then I need to show that $\varphi$ is a isometric isomorphim, i.e,

  1. $\varphi$ is linear.
  2. $\varphi$ is continuous.
  3. $\varphi(x)\in l_2(I)$.
  4. $\varphi$ is surjective.
  5. $\varphi$ is isometry.

My work:

  1. $\varphi$ is linear from linearity of inner product.
  2. Remember that $\|\varphi\|=\sup \{\| \varphi(x)\|_{l_2}: \|x\|=1\}$

Since $\{u_i:i\in I\}$ a orthonormal basis for Parseval equality we have for all $x\in\mathcal{H}$ $$\|\varphi(x)\|_{l_2}^2=\sum_{i\in I}|\langle x,u_i\rangle|^2=\|x\|^2=1.$$ Then $\|\varphi\|=1$ and $\varphi$ is continuous.

  1. Clearly, $\varphi$ is isometry for Parseval equality.

For 3, I need to use the Bessel inequality $\sum_{i\in I}|\langle x,u_i\rangle|^2\leq \|x\|^2$ but how do I use this to show what $\sum_{i\in I}|\langle x,u_i\rangle|^2$ is summable? I don´t see this.

For 4, I think I should use the Riesz Representation Theorem but again I don't know how to write the use of this theorem in the proo.

Thanks for your observations in my solutions of 1,2 and 5 and I hope can you help me in 3 and 4.

Thanks!

Luis Prado
  • 742
  • The answer here: https://math.stackexchange.com/questions/324538/separable-hilbert-space-have-a-countable-orthonormal-basis shows that $\mathcal{H}$ is actually separable. – Andrew Shedlock Oct 15 '20 at 19:29
  • In point 2. you already knew that $\sum|\langle x,u_i\rangle|$ equals to $|x|$, hence it is finite, which is the same as 'summable' for nonnegative numbers. – Berci Oct 15 '20 at 20:38

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