Ref: “Introduction to Hilbert spaces” by Debnath, Mikusinski.

Link to Stackexchange post: Link.

Let ${ (E, \lVert \ldots \rVert) }$ be a normed space.

Note that ${ (E, \lVert \ldots \rVert) }$ need not be complete.

Q) Can we construct a complete normed space ${ (\tilde{E}, \lVert \ldots \rVert _1) , }$ such that ${ (E, \lVert \ldots \rVert) }$ can be viewed as a dense subspace of ${ (\tilde{E}, \lVert \ldots \rVert _1) }$?

First, consider the fundamental reason ${ (E, \lVert \ldots \rVert) }$ can fail to be complete: It has a Cauchy sequence (a sequence where terms get close to each other) which doesn’t converge.

We are attempting to fix this by the above required construction.

Say two Cauchy sequences ${ (x _n ), (y _n) }$ “point to the same extended point” if ${ (x _n - y _n) \to 0 . }$

Q) Is the above relation an equivalence relation on all Cauchy sequences?

Note that the above relation is an equivalence relation on all Cauchy sequences in ${ E . }$

Let ${ \tilde{E} }$ be the set of all equivalence classes ${ [(x _n)] }$ where ${ (x _n) }$ is a Cauchy sequence in ${ E . }$

Note that ${ [(x _n)] }$ intuitively is the equivalence class of all Cauchy sequences which “point to the same extended point” as ${ (x _n) . }$ We intuitively treat it as an “extended point”.

Note that intuitively ${ \tilde{E} }$ is the set of all “extended points”.

Q) Is there a natural vector space structure on ${ \tilde{E} }$?

Note that

\[{ {\begin{aligned} &\, [(x _n)] + [(y _n)] := [(x _n + y _n)], \\ &\, \lambda [(x _n)] := [(\lambda x _n)] \end{aligned}} }\]

are well defined operations, and define a vector space structure on ${ \tilde{E} . }$

Q) Is there a natural normed space structure on ${ \tilde{E} }$?

Note that

\[{ \lVert [x _n] \rVert _1 := \lim _{n \to \infty} \lVert x _n \rVert }\]

is well defined, and defines a norm on ${ \tilde{E} . }$

Q) Can we view ${ (E, \lVert \ldots \rVert) }$ as a dense subspace of ${ (\tilde{E}, \lVert \ldots \rVert _1) }$?

Note that every element ${ x \in E }$ can be identified with ${ [(x, x, x, \ldots )] }$ in ${ \tilde{E} . }$

Note that the above identification preserves vector space structure and norm structure.

Hence ${ (E, \lVert \ldots \rVert) }$ can be viewed as a subspace of ${ (\tilde{E}, \lVert \ldots \rVert _1) . }$

Note that every element ${ [(x _n)] }$ in ${ \tilde{E} }$ is the limit of ${ x _1, x _2, x _3, \ldots }$ in ${ E . }$

Consider a point ${ [(x _n)] }$ in ${ \tilde{E} . }$ Note that

\[{ \lVert [(x _n)] - x _j \rVert _1 = \lim _{n \to \infty} \lVert x _n - x _j \rVert . }\]

Note that

\[{ \lVert [(x _n)] - x _j \rVert _1 \to 0 }\]

as ${ j \to \infty , }$ as needed.

Hence ${ (E, \lVert \ldots \rVert) }$ can be viewed as a dense subspace of ${ (\tilde{E}, \lVert \ldots \rVert _1) . }$

Q) Is ${ (\tilde{E}, \lVert \ldots \rVert _1) }$ complete?

Let ${ (X _n) }$ be a Cauchy sequence in ${ \tilde{E} . }$
We are to show it converges in ${ \tilde{E} . }$

Note that since ${ E }$ is dense in ${ \tilde{E}, }$ for every ${ n \in \mathbb{Z} _{> 0} }$ there is an ${ x _n \in E }$ with

\[{ \lVert X _n - x _n \rVert _1 < \frac{1}{n} . }\]

Note that

\[{ {\begin{aligned} &\, \lVert x _n - x _m \rVert _1 \\ \leq &\, \lVert x _n - X _n \rVert _1 + \lVert X _n - X _m \rVert _1 + \lVert X _m - x _m \rVert _1 \\ \leq &\, \lVert X _n - X _m \rVert _1 + \frac{1}{n} + \frac{1}{m} . \end{aligned}} }\]

Hence ${ (x _n) }$ is a Cauchy sequence in ${ E . }$

Hence consider the point ${ [(x _n)] }$ and the difference

\[{ \lVert X _n - [(x _n)] \rVert _1 . }\]

Note that

\[{ {\begin{aligned} &\, \lVert X _n - [(x _n)] \rVert _1 \\ \leq &\, \lVert X _n - x _n \rVert _1 + \lVert x _n - [(x _n)] \rVert _1 \\ \to &\, 0 . \end{aligned}} }\]

Hence ${ (X _n) }$ is convergent in ${ \tilde{E} , }$ as needed.

Hence we have constructed a complete normed space ${ (\tilde{E}, \lVert \ldots \rVert _1) , }$ such that ${ (E, \lVert \ldots \rVert) }$ can be viewed as a dense subspace of ${ (\tilde{E}, \lVert \ldots \rVert _1) , }$ as needed.