Sentence examples for a are compact from inspiring English sources

This is basically because the action spaces P k ( a ) are compact spaces and the utility functions are continuous with respect to the action profile.

(3.20) From [20] and [13], we know that all entries of the matrix operator A are compact, hence, we can easily see that the matrix operator A is compact in the weighted product space Y.

(3.33) From [20] and [13], we know that all entries of the matrix operator A are compact, hence, we can easily see that the matrix operator A is compact in the weighted product space Z.

Note that { x n } ⊂ ⋃ n = 1 + ∞ A n ∪ A, as well as A n and A are compact and A n → A, due to Lemma 2.7, { x n } or its subsequence converges to a point of A. Hence the right-hand side set of (5.1) is well-defined and nonempty.

(a) is compact ( is relatively compact).

(3) sup ( A ) is compact.

A is compact and continuous;   3.

(2) If A is compact, then A is totally bounded.  .

A is compact if and only if A is sequentially compact.

So, ϒ ( A ( B ) ) = 0. Therefore, A is compact.

Now, we prove that A is compact and continuous, respectively.