Normed Linear Spaces

If $K \subseteq \mathbb {R}$ is compact, show that $\left\| \cdot \right\|_{\max }$ is a norm on $C(K)$.

For $f \in C[a, b]$, define $\left\| f\right\|_{1} = \int_{a}^{b}\left|f\right| d m$. Show that this is a norm on $C[a, b]$. Also show that there is a number $c \geq 0$ for which

but there is no $c \geq 0$ for which

Let $X$ be the linear space of all polynomials with real coefficients defined on $\mathbb {R}$. For a polynomial $p$, define $\left\| p\right\|$ to be the sum of the absolute values of the coefficients. Is this a norm?

For $f \in L^{1}[a, b]$, define $\left\| f\right\| = \int_{a}^{b} x^{2}\left|f(x)\right| d x$. Show that this is a norm on $L^{1}[a, b]$.

For $f \in L^{\infty }(E)$, show that $\left\| f\right\|_{\infty }$ is the smallest essential upper bound for $f$ and

For $0 < p < 1$, define $L^{p}(E)$ to be the collection of measurable functions $f: E \rightarrow \overline{\mathbb {R}}$ for which $\int_{E}\left|f\right|^{p} d m < \infty$. Show that $L^{p}(E)$ is a linear space.

Show that in a normed linear space, a convergent sequence is bounded.

State and prove the linearity property of convergent sequences in a normed linear space.