When are Linear functions on normed spaces continuous? And what Properties do they have? In Analysis and Linear Algebra classes one always talks about linear function and continuous function somewhat separately... so that let me to:
Suppose $(V,\left\lVert \cdot \right\rVert_V)$ and $(W,\left\lVert \cdot \right\rVert_W)$ are a normed vector space and $f:V \rightarrow W$ is an linear function. Since $V$ and $W$ are normed we can talk about the topology of $V$ and $W$ given by the open sets $B_V(x,r)=\{y \in V \mid \left\lVert x-y \right\rVert_V < r\}$ and  $B_W(x,r)=\{y \in W \mid \left\lVert x-y \right\rVert_W < r\}.$ For the continuity of $f$ we say that $f$ is continous at $x_0$ if:
$$\forall \varepsilon > 0  \ \forall x \in V \ \exists \delta(\varepsilon,x):\left\lVert x_0-x \right\rVert_V< \delta \Rightarrow \left\lVert f(x_0)-f(x) \right\rVert_W < \varepsilon.$$ Since $f$ linear we get from above $\left\lVert x_0-x \right\rVert_V< \delta \Rightarrow \left\lVert f(x_0-x) \right\rVert_W < \varepsilon$. So from that I see that the identify function is continuous, and orthogonal function as-well, since $\left\lVert v \right\rVert_V = \left\lVert f(v) \right\rVert_W$ thus set $\delta = \varepsilon$ (its even uniformal continuous).

My Question: Does $f$ need any additional property to be continuous or is linearity enought?  What kind of property are sufficient ? and if $f$ is linear and continous what does that say about $f$? For example the orthogonality above. And last but not least, does the dimension matter, are finite and infinite case the different?


My Backround: Undergrade maths student. I know some linear Algebra asswell as Analysis and Topology but nothing yet about functional Analysis

 A: Let $T\in\mathcal{L}(X,Y)$

*

*$T$ is continuous.


*$T$ is continuous at $x_0\in X$


*$T$ is continuous at $0$


*$\exists M>0$ such that $\forall x\in X$ $\|Tx\|\le M\|x\|$


*$T(B[0, 1]) $ is bounded.


*$T$ map bounded set to bounded set.
All the statements $(1) \to (6) $ are equivalent.
Examples: (continuous)

*

*$\operatorname{Tr}: \mathcal{M_n(\Bbb{K}) }\to K$ defined by $$\operatorname{Tr}(A) =\sum_{i=1}^{n} a_{ii}$$


*$Id :( C[0, 1], \|.\|_{\infty})\to ( C[0, 1], \|.\|_1)$
$$Id(f) =f$$


*$I:(\mathcal{R}[a, b], \|•\|) \to \Bbb{R}$
$$I(f) =\int_{a}^{b}f(t) dt$$

Examples: (not continuous)

*

*$Id :( C[0, 1], \|.\|_{\infty})\to ( C[0, 1], \|.\|_1)$
$$Id(f) =f$$


*$T_0 :( C^1 [-\pi, \pi], \|.\|_{\infty})\to \Bbb{R}$
$$T_0(f) =f'(0) $$

All the above discontinuous linear map defined on a infinite dimensional normed linear space!
Another condition (sufficient but not necessary) for continuity $\dim(X) <\infty$
All linear maps that we have already studied in Linear algebra (finite dimensional analysis) is continuous!

To check the continuity of a linear map we only need to check whether it is continuous at $0$ or not,  locally continuous iff globally continuous ( sequence gives an easy characterization) , an ordinary map doesn't enjoy this property. For an example
$f:\Bbb{R}\to \Bbb{R}$
$$f(x)=\begin{cases}x&x\in\Bbb{Q}\\0&\text{otherwise}\end{cases}$$
continuous only at $0$ .
For a linear map either it is continuous on the whole domain or it is discontinuous on the whole domain.

Linear maps defined between two normed linear spaces are just special, I like very much!  (Lost in space)
