The term 'euclidean space' means a space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional
The question doesn't make sense, or alternatively it is true by definition. A Hilbert Space is a complete inner product space - complete in the metric induced by the norm defined by the inner product...
Get a watter bottle. Dring a large gulp. Turn it upside down add that bubble is air.
Try to run really fast through a wall.
There is no mathematical proof that space is infinite. All we know is that there is an expanding limit to what we can see.