# Operator norm $( \ell_2 \to \ell_1)$

Let $$X$$ be a ﬁnite dimensional normed vector space and $$Y$$ an arbitrary normed vector space. $$T:X→Y$$.

I want to calculate $$\|T\|$$ for where $$X = K^n$$, equipped with the Euclidean norm $$\|\cdot\|_2$$, $$Y := \ell_1(\mathbb{N})$$ and $$Tx := (x_1,\ldots,x_n,0,0,\ldots) \in \ell_1(\mathbb{N})$$, for all $$x = (x_1,\ldots,x_n) \in K^n$$.

I do not know how to continue $$||T∥_2 = \sup \limits_{x \neq 0} \frac{∥Tx∥_1}{∥x∥_2} = \sup \limits_{x \neq 0} \frac{∥(x_1,…,x_n,0,0,…)∥_1}{∥(x_1,…,x_n)∥_2} = \sup \limits_{x \neq 0} \frac{|x_1|+…+|x_n|}{(|x_1|^2+…+|x_n|^2)^{\frac{1}{2}}}= ?$$

• I don't think that it's a good approach, but you can use the fact that $||T|| = \sup_{x: ||x||_{2} = 1} {||Tx||_{1}}$ and apply Lagrange multipliers then. In other words you shall find $|x_{1}| + \ldots + |x_{n}| \rightarrow \text{max}$, while $|x_{1}|^{2} + \ldots |x_{n}|^{2} = 1$ – hyperkahler Dec 16 '18 at 20:30
• I am tried around a lot. I have never worked with Lagrange multiplier, but $||T∥_2 = \sup \limits_{∥x∥_2 = 1} ∥Tx ||_1$. – Anna Schmitz Dec 16 '18 at 20:38

## 2 Answers

I will elaborate on my comment above

Given an operator $$T: X \rightarrow Y$$ where $$X = \mathbb{K}^{n}$$ $$Y = l^{1}(\mathbb{N})$$ its norm is given by $$||T||_{op} = \sup_{x \neq 0}{\frac{||Tx||_{1}}{||x||_{2}}} = \sup_{||x||_{2} \leq 1}{||Tx||_{1}} = \sup_{||x||_{2} = 1}{||Tx||_{1}}$$

So we have to maximize $$||Tx||_{1} = |x_{1}| + \ldots + |x_{n}|$$ given $$||x||_{2} := (x_{1}^{2} + \ldots + x_{n}^{2})^{\frac{1}{2}} = 1$$

Let $$t_{i} := |x_{i}|$$, then our problem reformulates as follows: $$t_{1} + t_{2} + \ldots + t_{n} \rightarrow \text{max}$$ $$t_{1}^{2} + t_{2}^{2} + \ldots + t_{n}^{2} = 1$$ $$t_{i} \geq 0, \ \forall i = 1, \ldots, n$$

The Lagrangian for the problem is: $$L = (t_{1} + \ldots + t_{n}) - \lambda (t_{1}^{2} + \ldots + t_{n}^{2} - 1)$$ and $$\lambda$$ stays for the lagrange multiplier.

The necessary extremum condition implies: $$\frac{\partial L}{\partial t_{i}} := 1 - \lambda(2t_{i}) = 0$$ thus $$t_{i} = \frac{1}{2 \lambda}$$. Since $$t_{i} \geq 0$$ it follows that $$\lambda > 0$$.

The stationary point is $$x = (\frac{1}{2 \lambda}, \ldots, \frac{1}{2 \lambda})$$ and we shall find the lambda such that the solution satisfy the second condition in the problem above.

$$||x||_{2} = 1$$ is equivalent to $$n \cdot \frac{1}{4 \lambda^{2}} = 1$$ from which we get $$\lambda = \frac{\sqrt{n}}{2}$$ (note that we shall choose it positive due to the restrictions mentioned above)

Thus $$t_{i} = \frac{1}{\sqrt{n}}$$ and the maximum equals $$\text{max} = n \cdot \frac{1}{\sqrt{n}} = \sqrt{n}$$

• Hi, how does $n \cdot 1/n = \sqrt{n}$ ? Also, nice alternative thinking but that's a bit off the road of functinal analysis. – Rebellos Dec 16 '18 at 21:16
• @Rebellos i've made a mistake, please, check the answer for the updates – hyperkahler Dec 16 '18 at 21:25

Using Cauchy-Schwarz inequality, we have $$\sum_{i=1}^n\left\lvert x_i\right\rvert=\sum_{i=1}^n\left\lvert x_i\right\rvert\cdot 1\leqslant \left(\sum_{i=1}^nx_i^2\right)^{1/2}\left(\sum_{i=1}^n1\right)^{1/2}=\sqrt n\left(\sum_{i=1}^nx_i^2\right)^{1/2}$$ hence $$\left\lVert T\right\rVert\leqslant \sqrt n$$. For the opposite inequality, look at the case where $$x_i=1$$ for all $$i\in\{1,\dots,n\}$$.