Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying space $X$ has the same topology with respect to either norm. Maybe it preserves even more properties than this too.

But I'm finding it very difficult to use this property when doing proofs or problems because although it's very simple to state, I don't immediately see what it is saying. In comparison, when you define 'equivalence' in other settings, like in the definition of an isomorphism of abelian groups or a continuous map, it's very clear that a certain operation or object is being preserved as you pass across a map.

To be concrete, here are my questions:

(1) Is there another way to characterize when norms are equivalent that might provide more intuition for what it says about is being preserved

and (2) is there a way to show that this definition is the one you want by starting with something more fundamental (like saying that the norms induce the same topology) and then proving that it's equivalent to the stated definition?

Any intuition for the definition would be helpful for either of these questions.

Solution 1:

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.

Solution 2:

Maybe it will be useful to consider an example of two norms $F$ and $G$ of a vector space $X$ not being equivalent to each other. What it means is that at least one of the quantities $\sup\limits_{x \in X}\frac{F(x)}{G(x)}$ or $\sup\limits_{x \in X}\frac{G(x)}{F(x)}$ is unbounded, i.e. there is a sequence $(x_n)_{n \geq 0}$ of vectors in the space such that $\frac{F(x_n)}{G(x_n)}$ or $\frac{G(x_n)}{F(x_n)}$ diverges to $+\infty$ when $n \to \infty$.

Intuitively, if a norm $G$ is equivalent to a norm $F$, $G$ neither stretches too much, nor shortens too much the lengths of the vectors already assigned by $F$. The "new" $G-$length of every vector is within a bound of its "old" $F-$length: $\forall x \in X, \quad aF(x) \leq G(x) \leq bF(x)$. There are not parts of the space stretched or shrunk arbitrarily much.

For an example of two nonequivalent norms, consider the space $C^2([0,1],\mathbb{R})$ of real twice continuously differentiable functions defined on $[0,1]$. Define the two norms $F(x(t)) = \sup\limits_{t \in [0,1]}|x(t)|$ and $G(x(t)) = |x(0)| + \sup\limits_{t \in [0,1]}|x'(t)|$. Now, if we look at the sequence $x_n(t) = \sin(n\pi t)$, then the $F-$norm of every $x_n$ is $1$ while $G(x_n) = n\pi$. The unit $F-$sphere is torn away to infinity when the $G$ norm is used.

Solution 3:

Suppose you have a sequence which converges in $G $. The lower bound implies it converges in $F $ to the same limit.

Suppose you have a sequence which does not converge in $G $. The upper bound implies it does not converge in $F $.

That's all you need, since metric spaces are sequential spaces.

Solution 4:

You can start with "the norms induce the same topology". Then use the fact that a linear transformation is continuous if and only if it is bounded. And this is one of your inequalities. For the other direction, use the inverse of that linear transformation.