Why a linear map on $R$ is automatically continuous, but a linear map between normed linear space has to be bounded to become continuous?

Question

From basic calculus, we know that if we define a linear function $f(x)=Ax$ with $A\neq 0$ and $x\in R$, then $f(x)$ is automatically continuous in the sense of $\underset{x_n\rightarrow x}{lim} f(x_n)=f(x)$ for any $x$ (and $f(x)$ is also automatically ``bounded'' in the sense of $|f(x)|\leq|A||x|$ for any $x\in R$ ). However, in functional analysis (see the notes at https://uwaterloo.ca/scholar/sites/ca.scholar/files/g6tran/files/amath731_lecture10.pdf), a linear operator (or linear mapping)$L:X\rightarrow Y$ is not necessarily continuous in the sense of $\underset{x_n\rightarrow x}{lim}L(x_n)=L(x)$. Neither is it necessarily bounded in the sense of there exists $c$ such that $||Lx||_Y\leq c||x||_X$ for any $x\in X$. That's why we must first define a bounded linear operator which satisfies $||Lx||_Y\leq c||x||_X$ for any $x\in X$, and then establish its equivalence to continuity of $L$.

My question is, intuitively, as we generalize a linear mapping $f:R\rightarrow R$ to a linear mapping $L:X\rightarrow Y$, what happens so that $L$ is not necessarily bounded (in the sense of $||Lx||_Y\leq c||x||_X$ for any $x\in X$) or continuous?
Some example for linear operator $L$ that is not bounded or not continuous might be helpful. Thanks!

Answer 1

A classical example of a linear but unbounded operator is the derivative operator. Define $$D : \big( C^1[0,1], ||\cdot||_{\infty} \big) \to \big( C[0,1], ||\cdot||_{\infty} \big)$$ as an operator where $C^1[0,1]$ denotes the set of continuously differentiable functions over $[0,1]$. Define $f_n(x) = x^n$. We have $||f_n||= 1$, but $||Df_n|| = n$ which tends to infinity.

Answer 2

The equivalence between boundedness and continuity at the origin is true in arbitrary normed space. In finite dimension, linear maps $L : \mathbb{R}^p \to \mathbb{R}^n$ have $(Lx)_{i} = \sum_{j = 1}^{p}L_{ij}x_j$, which is obviously continuous.

An important example is to take a space $X$ with different norms $\|\cdot\|_1$ and $\|\cdot\|_2$ and investigate the identity map $I : (X, \|\cdot\|_1) \to (X, \|\cdot\|_2)$. Continuity is equivalent to $\|x\|_1 \leq C\|x\|_2$