When I studied on norms: vectors are defined by : $x\in R^n$.
Many books said $x$ is a column vector. (Also, define inner product $x^tx$.)
What I am curious about is : $||x|| =||x^t||$? (any) norms of vector and its transpose are the same.
We should see $||x^t||$ or calculate as like a column vector? or is it a matrix, and calculate like induced norm?
The elements $x\in{\mathbb R}^n$ are just $n$-tuples $(x_1,x_2,\ldots, x_n)$ and, a priori, neither row nor column vectors. If you use them for doing euclidean geometry, say, studying ellipses or spirals, then you usually use the norm $$|x|:=\sqrt{x_1^2+\ldots+x_n^2}\ .\tag{1}$$ When you are doing in fact linear algebra then you consider the points of the given "base space" $X:={\mathbb R}^n$ as column vectors and set up the standard matrix calculus. In this environment you sometimes also use norms, e.g., the above norm $(1)$, which then is denoted $\|x\|_2$. In some cases you use other norms, e.g., $$\|x\|_1:=\sum_{k=1}^n |x_k|,\qquad{\rm or}\qquad \|x\|_\infty:=\max_{1\leq k\leq n}|x_k|\ .$$ These norms refer to the "base space" $X$. To this space is associated its dual $X^*$ containing the linear functionals $\phi: \>X\to{\mathbb R}$. Such a functional appears as $$\phi(x)=\sum_{k=1}^n a_k x_k\ ,$$ so that "datawise" this $\phi$ is again encoded by an $n$-tuple $(a_1,\ldots,a_n)=:a$. In matrix calculus this $n$-tuple $a$ is then a row vector (and is sometimes written $a^{\rm t}$, or similar). When we have some norm $\|\cdot\|$ on the base space $X$ the $\phi$ has automatically its functional norm $$\|\phi\|:=\sup_{x\ne0}{|\phi(x)|\over\|x\|}\ .$$ When the norm in $X$ is the euclidean norm $\|\cdot\|_2$ then $\|\phi\|=\|a\|_2$ again, but when the norm in $X$ is $\|\cdot\|_1$ then the "associated" norm in $X^*$ is $\|\phi\|=\|a\|_\infty$. (The proof is an exercise.)
