# What does this norm $|| \cdot ||_{{C^0} (K) }$ mean?

$$|| A ||_{C^0 (K)}$$ Here $A$ is $n \times n$ Hermitian, Positive definite matrix, and $K \in \mathbb R^n$.

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Hi Ashuley! Could you give more context here? Is $K$ just a point in $\mathbb{R}^n$, or is a compact subset? Are there functions anywhere? Usually $C^0$ denotes the space of continuous functions on some set. – froggie May 13 '12 at 17:17
@froggie oh i'm sorry. In fact, $A$ is consists of functions in $C_b^1$. $C_b^1$ means that the collection of 1 time diff'ble functions with a bounded derivative. – Misaj May 13 '12 at 17:24
$K$ means that a truncated cone in $R^n$. – Misaj May 13 '12 at 17:25
In that case, my guess is that $$\|A\|_{C^0(K)} = \max_{i,j}\sup_{p\in K}|A_{ij}(p)|.$$ Here $A_{ij}$ is the $(i,j)$th entry of $A$. – froggie May 13 '12 at 17:26
@froggie Oh thank you. There is not any explanation in the book, but I think you're right. – Misaj May 13 '12 at 17:37

As the notations may suggest, $K$ is compact and $A(x)$ is a matrix when $x\in K$ and the entries are continuous functions. Then $$\lVert A\rVert_{C^0(K)}=\max_{r,c}\sup\{A_{r,c}(x),x\in K\}.$$

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