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## Main Question or Discussion Point

Correct me if I'm wrong here but it is my understanding that vector spaces are given structure such as inner products, because it allows us to use these structured vector spaces to describe and analyse physical things with them.

So physical properties such as 'distance' cannot be analysed in a general vector space but have analogies in a vector space [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}^3[/itex] over the Real Numbers in an inner product space.

Assuming my understanding is correct I'm a bit confused at the connection/relationship between Metric spaces, Inner product spaces and Normed linear spaces and which are more general and which are subsets of which.

My understanding is that Metric spaces are subsets of Inner product spaces which are subsets of Normed linear spaces.

An Inner product space is a normed linear space with a specific definition of the norm, namely [itex]||a|| = \sqrt{\langle a|a\rangle}[/itex] whereas a normed linear space is

Is this correct?

So physical properties such as 'distance' cannot be analysed in a general vector space but have analogies in a vector space [itex]\mathbb{R}^2[/itex] and [itex]\mathbb{R}^3[/itex] over the Real Numbers in an inner product space.

Assuming my understanding is correct I'm a bit confused at the connection/relationship between Metric spaces, Inner product spaces and Normed linear spaces and which are more general and which are subsets of which.

My understanding is that Metric spaces are subsets of Inner product spaces which are subsets of Normed linear spaces.

An Inner product space is a normed linear space with a specific definition of the norm, namely [itex]||a|| = \sqrt{\langle a|a\rangle}[/itex] whereas a normed linear space is

**any**vector space together with a function that obeys the properties of a norm.Is this correct?