The expansion of the Universe should not be thought as if matter expands quickly in an existing space; it is space itself that is expanding. This is a confusing topic; read the Wikipedia article on "metric expansion of space" for a brief introduction.

The assumptions of a metric space except for symmetry.

The assumptions of a metric space except for symmetry.

A metric on a set is complete if every Cauchy sequence in the corresponding metric space they form converges to a point of the set in question. The metric space itself is called a complete metric space.

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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 over the space.

In other words an inner product space is a vector space with an inner product defined on it.

An inner product then defines a norm on the space, and every norm on a space induces a metric.

A Hilbert Space is thus also a complete metric space, simply where the metric is induced by the inner product.