Making Spaces
In quantum mechanics a lot of emphasis is placed on the concept of vector spaces. One of the key tools is the ability to construct new vector spaces out of existing ones. However, very often authors construct new vector spaces without explicitly saying what they have done, and the result can be confusing. In this post I am going to attempt to summarize all the methods I have seen for constructing new vector spaces out of old, and point out where they are used in quantum mechanics. The building blocks What: Hilbert Spaces Why: To represent superpositions of classical states The building blocks are always Hilbert spaces. These are vector spaces over the complex numbers $\mathbb{C}$, with inner products and limits. The pair $(V, \langle\cdot\lvert\cdot\rangle)$ is a Hilbert space if $V$ is a vector space over $\mathbb{C}$ $v\mapsto\langle u\lvert v\rangle$ is linear map $V\mapsto\mathbb{C}$ for any $u$ in $V$ $\langle u\lvert v\rangle = \overline{\langle v\lvert u\rang