# Linear transformation

A norm of linear transformation $$\Lambda : X \rightarrow Y$$ is defined by $$\| \Lambda \| = \sup \{ \| \Lambda(x) \|:x \in X, \|x\| \le 1 \}$$. We can give a norm to a space or a set. A norm determines the size of a vector in the function space. The way of measure the size of a vector gives important properties of the space like boundness, completeness or orthogonality. We can also select a set ($$L^{1}(\mu)$$) to have a certain properties like boundness and completeness in a specific norm like $$\| * \|_2$$ and so on select $$C(X)$$ for $$L^{\infty}$$.

The linear transformation $$\Lambda$$ can compose a normed space $$\{\Lambda\}$$. The space of linear transformation can have properties like boundness, completeness. The Banach-Steinhaus theorem states the limit of linear transformation between Banach spaces which a set of bounded linear transformation, $$\{\Lambda_{\alpha}\}$$, $$\alpha \in A$$ with Banach space in both domain and codomain has either bounded norm or unbounded supremum $$\| \Lambda_{\alpha} \|$$ for all some dense $$G_\delta$$ in domain. The limit of linear transformation space can be applied in convergence of differentiation and integration of simple function and Fourier transformation because those are series of linear transformation.

##### Jun Kang
###### Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.