Constructing Unbounded Linear Maps Suppose that $X$ is an infinite-dimensional normed vector space over $\mathbb C$ (that is, the cardinality of any of its Hamel bases is infinite) and let $Y$ be another, nontrivial normed vector space over $\mathbb C$ (that is, $Y$ contains other elements beyond the zero vector). I am wondering about the following questions:


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*If $X$ is incomplete (that is, there exists at least one Cauchy sequence in $X$ that fails to converge in $X$), how can one construct a linear function from $X$ to $Y$ that is unbounded (or, equivalently, not continuous) explicitly, i.e., without using the axiom of choice or other results hinging on it (e.g., the Hahn–Banach theorem)?

*If $X$ is complete, how can one construct an incomplete subspace $X_0$ of it such that the completion of $X_0$ is precisely $X$?


I appreciate any hints or references to standard results.

Proving the existence of Schauder bases sometimes relies on the axiom of choice, but many important normed spaces have explicit Schauder bases.
Suppose we know a (countable) Schauder basis for $X$, call it $\{x_i\}$. The subspace $X_0$ for which $\{x_i\}$ is a Hamel basis is incomplete. Explicitly $X_0=\{\sum_{j=1}^Nc_jx_{i_j}:N\in\mathbb{N},c_j\in\mathbb{C}\}$.
To show $X_0$ is incomplete, notice that the sequence $\left\{\sum_{j=1}^Nx_j2^{-j}\right\}_{N=1}^\infty$ is Cauchy but does not converge in $X_0$. It shouldn't be hard to show $X$ is the completion of $X_0$ (basically that's how a Schauder basis is defined).
You can construct an unbounded linear functional $f:X_0\to\mathbb{C}$ easily: $f(\sum_{j=1}^Nc_jx_{i_j})=\sum_{j=1}^Ni_jc_j$.
To show $f$ is unbounded, assume $\|x_i\|=1$ for each $i$. Then $f(x_i)=i$ and so $\{x_i\}$ is a sequence on the unit sphere with an unbounded image.
If $0\neq y\in Y$, you can define an unbounded linear function $g:X_0\to Y$ by $g(x)=yf(x)$.
