PROBLEM
Answer
Equivalent norms define the same uniform vector space.
Explanation
Completeness is a concept by uniform vector spaces.
Demonstration
Given the real line $\mathbb{R}$.
Consider the metrics:
$$d(x,y):=|y-x|$$
$$d'(x,y):=\arctan|y-x|$$
Then one obtains:
$$\mathcal{N}=\mathcal{N}'\quad\mathcal{U}\neq\mathcal{U}$$
Concluding problem.
PROOF
Identification
Given normed spaces $\Omega$ and $\Omega'$.
Regard the category:
$$\mathrm{UVS}:\quad\mathrm{Hom}(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_U(\Omega,\Omega')$$
Identification:
$$\Phi:\Omega\leftrightarrow\Omega':x\mapsto x$$
Linearity follows:
$$\Phi(x+y)=x+y=x+'y=\Phi(x)+'\Phi(y)$$
$$\Phi(\lambda\cdot x)=\lambda\cdot x=\lambda\cdot'x=\lambda\cdot'\Phi(x)$$
But by the below:
$$\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_U(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_L(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{B}(\Omega,\Omega')$$
Explicitely that is:
$$\|x\|'=\|\Phi x\|'\leq\|\Phi\|\cdot\|x\|$$
$$\|x'\|=\|\Phi^{-1}x'\|\leq\|\Phi^{-1}\|\cdot\|x'\|'$$
Concluding proof.
CATEGORIES
Topological Vector Spaces
Note for linear maps:
$$\Phi\in\mathcal{L}(\Omega,\Omega')\implies\Phi^{-1}\in\mathcal{L}(\Omega',\Omega')$$
Continuous at zero:
$$\mathcal{C}_0(\Omega,\Omega'):=\{\Phi:\Omega\to\Omega:\Phi^{-1}(\mathcal{N}_{\Phi0})\subseteq\mathcal{N}_0\}$$
It holds equality:
$$\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_0(\Omega,\Omega')$$
Homomorphisms:
$$\mathrm{Hom}(\Omega,\Omega'):=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}(\Omega,\Omega')$$
Isomorphic spaces:
$$\Omega\cong\Omega':\iff\Phi:\Omega\leftrightarrow\Omega':\quad\Phi(\mathcal{N})=\mathcal{N}'$$
$$\Phi(x+y)=\Phi(x)+'\Phi(y)\quad((x,y)\in\Omega\times\Omega)$$
$$\Phi(\lambda\cdot x)=\lambda\cdot'\Phi(x)\quad((x,\lambda)\in\Omega\times\mathbb{C})$$
Basic entourages:
$$B_N:=\{(x,y):(y-x)\in N\}\subseteq\Omega\times\Omega$$
Uniform structure:
$$\mathcal{U}:=\uparrow\{B_N: N\in\mathcal{N}_0\}$$
Going first step up:
Uniform Vector Spaces
Neighborhoods:
$$\mathcal{N}_z:=\{U[z]:U\in\mathcal{U}\}$$
Uniform maps:
$$\mathcal{C}_U(\Omega,\Omega'):=\{\Phi:\Omega\to\Omega':\Phi^{-1}(\mathcal{U}')\subseteq\mathcal{U}\}$$
It holds equality:
$$\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_U(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}(\Omega,\Omega')$$
Homomorphisms:
$$\mathrm{Hom}(\Omega,\Omega'):=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_U(\Omega,\Omega')$$
Isomorphic spaces:
$$\Omega\cong\Omega':\iff\Phi:\Omega\leftrightarrow\Omega':\quad\Phi(\mathcal{U})=\mathcal{U}'$$
$$\Phi(x+y)=\Phi(x)+'\Phi(y)\quad((x,y)\in\Omega\times\Omega)$$
$$\Phi(\lambda\cdot x)=\lambda\cdot'\Phi(x)\quad((x,\lambda)\in\Omega\times\mathbb{C})$$
Suppose one finds:
$$\text{Locally Convex Base}$$
Induced seminorms:
$$\mu_U(x):=\inf\{r\geq0:x\in rU\}$$
Going next step up:
Locally Convex Spaces
Basic entourages:
$$B_{\mu\varepsilon}:=\{(x,y):\mu(y-x)<\varepsilon\}\subseteq\Omega\times\Omega$$
Uniform structure:
$$\mathcal{U}:=\uparrow\{B_{\mu\varepsilon}:\mu\in\mathcal{S},\varepsilon>0\}$$
Isomorphic spaces:
$$\Omega\cong\Omega':\iff\Phi:\Omega\leftrightarrow\Omega':\quad\Phi(\mathcal{S})=\mathcal{S}'$$
$$\Phi(x+y)=\Phi(x)+'\Phi(y)\quad((x,y)\in\Omega\times\Omega)$$
$$\Phi(\lambda\cdot x)=\lambda\cdot'\Phi(x)\quad((x,\lambda)\in\Omega\times\mathbb{C})$$
Suppose one finds:
$$\text{Countable Base}$$
Induced Metric:
$$d(x,y):=\sum_{k=1}^\infty\frac{1}{2^k}\frac{\sigma_k(y-x)}{1+\sigma(y-x)}$$
Going next step up:
Metrizable Vector Space
Induced seminorm:
$$\mu(x):=d(x,0)=d(0,x)\geq0$$
Lipschitz maps:
$$\mathcal{C}_L(\Omega,\Omega'):=\{\Phi:\Omega\to\Omega':d(\Phi\cdot,\Phi\cdot)'\leq L_\Phi d(\cdot,\cdot)\}$$
It holds equality:
$$\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_L(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_U(\Omega,\Omega')$$
Isometric maps:
$$\mathcal{I}(\Omega,\Omega'):=\{\Phi:\Omega\to\Omega':d(\Phi\cdot\Phi\cdot)'=d(\cdot,\cdot)\}$$
Homomorphisms:
$$\mathrm{Hom}(\Omega,\Omega'):=\mathcal{L}(\Omega,\Omega')\cap\mathcal{I}(\Omega,\Omega')$$
Isomorphic spaces:
$$\Omega\cong\Omega':\iff\Phi:\Omega\leftrightarrow\Omega':\quad d(\Phi\cdot,\Phi\cdot)'=d(\cdot,\cdot)$$
$$\Phi(x+y)=\Phi(x)+'\Phi(y)\quad((x,y)\in\Omega\times\Omega)$$
$$\Phi(\lambda\cdot x)=\lambda\cdot'\Phi(x)\quad((x,\lambda)\in\Omega\times\mathbb{C})$$
By construction:
$$d(x+a,y+a)=d(x,y)\quad(a\in\Omega)$$
Suppose one has:
$$d(\lambda x,\lambda y)=|\lambda|d(x,y)\quad(\lambda\in\mathbb{C})$$
Induced norm:
$$\|x\|:=d(x,0)\geq0$$
Going next step up:
Normed Spaces
Induced metric:
$$d(x,y):=\|y-x\|\geq0$$
Lipschitz maps:
$$\mathcal{B}(\Omega,\Omega'):=\{\Phi:\Omega\to\Omega':\|\Phi\cdot\|'\leq\|\Phi\|\cdot\|\cdot\|\}$$
It holds equality:
$$\mathcal{L}(\Omega,\Omega')\cap\mathcal{B}(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{C}_L(\Omega,\Omega')$$
Isomorphic spaces:
$$\Omega\cong\Omega':\iff\Phi:\Omega\leftrightarrow\Omega':\quad\|\Phi(\cdot)\|'=\|\cdot\|$$
$$\Phi(x+y)=\Phi(x)+'\Phi(y)\quad((x,y)\in\Omega\times\Omega)$$
$$\Phi(\lambda\cdot x)=\lambda\cdot'\Phi(x)\quad((x,\lambda)\in\Omega\times\mathbb{C})$$
Suppose one has:
$$\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2$$
Induced scalar product:
$$\langle x,y\rangle:=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha\|x+i^\alpha y\|$$
Going final step up:
Hilbert Spaces
Induced norm:
$$\|x\|^2:=\langle x,x\rangle\geq0$$
Orthogonal maps:
$$\mathcal{O}(\Omega,\Omega'):=\{\Phi:\Omega\to\Omega':\langle\Phi\cdot,\Phi\cdot\rangle'=\langle\cdot,\cdot\rangle\}$$
It holds equality:
$$\mathcal{L}(\Omega,\Omega')\cap\mathcal{O}(\Omega,\Omega')=\mathcal{L}(\Omega,\Omega')\cap\mathcal{I}(\Omega,\Omega')$$
Homomorphisms:
$$\mathrm{Hom}(\Omega,\Omega'):=\mathcal{L}(\Omega,\Omega')\cap\mathcal{O}(\Omega,\Omega')$$
Isomorphic spaces:
$$\Omega\cong\Omega':\iff\Phi:\Omega\leftrightarrow\Omega':\quad \langle\Phi\cdot,\Phi\cdot\rangle'=\langle\cdot,\cdot\rangle$$
$$\Phi(x+y)=\Phi(x)+'\Phi(y)\quad((x,y)\in\Omega\times\Omega)$$
$$\Phi(\lambda\cdot x)=\lambda\cdot'\Phi(x)\quad((x,\lambda)\in\Omega\times\mathbb{C})$$
Concluding categories.