Prove that Hilbert Space is a Metric Space?

Answer 1

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.