Hahn-Banach theorem

Theorem, ($\mathbb{R}$ version)

Let $\mathcal{X}$ be an $\mathbb{R}$-vector space. Suppose $q : \mathcal{X} \to \mathbb{R}$ is a quasi-seminorm. Suppose also we are given a linear subspace $\mathcal{Y} \subset \mathcal{X}$ and a linear map $\phi : \mathcal{Y} \to \mathbb{R}$, such that $$\phi(y) \leq q(y), \quad \forall y \in \mathcal{Y}$$ Then there exists a linear map $\psi: \mathcal{X} \to \mathbb{R}$ such that $\psi |_\mathcal{Y} = \phi$ and $\psi(x) \leq q(x)$ for all $x \in \mathcal{X}$.

Theorem, (normed linear spaces)

#incomplete

References

1. https://www.ucl.ac.uk/~ucahad0/3103_handout_6.pdf
2. https://www.math.ksu.edu/~nagy/real-an/ap-e-h-b.pdf