Complex Vector Space with Dot Product is Hilbert Space

Theorem

Let $\C^d$ be a complex vector space with $d$ dimensions.

Let $\innerprod \cdot \cdot$ be the dot product on $\C^d$.

There is believed to be a mistake here, possibly a typo. In particular: This dot product is only defined on $\C$, not on $\C^d$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Then $\C^d$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space.

Proof

Let us explore the inner product norm of $\innerprod \cdot \cdot$:

$\ds \norm z = \sqrt{ \sum_{i \mathop = 1}^d x_i^2 + y_i^2 }$

and subsequently the metric induced by $\norm \cdot$:

$\ds \map d {z, z'} = \norm {z - z'} = \sqrt{ \sum_{i \mathop = 1}^d \paren{ x_i - x'_i }^2 + \paren{ y_i - y'_i }^2 }$

This theorem requires a proof. In particular: Elementary results around $\C^d$ seem to be missing You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples