Suppose $X$ is a vector space and $\mathcal{P}$ a family of seimorms on $X$. Let $\mathcal{T}$ be the topology on $X$ that has as a subbase the sets $\{x: p(x-x_0)<\varepsilon\}$. So the base of $X$ has the form $$\cap_{j=1}^n\{x\in X: p_j(x-x_0)<\varepsilon_j\}.$$
This statment is from Page 99 of the book "A Course in Functional Analysis" by J.B. Conway. However, this form of the base is different from which is in the book "Fundamentals of the theory of operator algebras" of R.V. Kadison. In Page 17 of Kadison's book, the Theorem 1.2.6 introduced the form of base of $X$ is $$\{x\in X: p_j(x-x_0)<\varepsilon~(j=1,...,m)\}.$$
In Conway's book there are n $\varepsilon_j$ and, in Kadison's book, there is only one $\varepsilon$. Are these two forms same?