1. if A is nonempty subset of a linear space, then the \(\hspace{1.0cm}\) of all convex sets containing A gives you the convex hull of A denoted by co A.

universal

complement

union

—>> intersection

2. Let \(\textbf{K} \cdot \textbf{K}_1\) and \(\textbf{K} \cdot \textbf{K}_2\) be two norms defined on a linear space X. \(\textbf{K} \cdot \textbf{K}_1\) and \(\textbf{K} \cdot \textbf{K}_2\) are equivalent if there exists constants \(a, b > 0\) such that \(\hspace{1.0cm}\) for all \(x \in X\).

\(a \textbf{K} \times \textbf{K}_1 \geq 0\)

\(a \textbf{K} \times \textbf{K}_1 \geq \textbf{K} \times \textbf{K}_2\)

\(a \textbf{K} \times \textbf{K}_1 \leq \textbf{K} \times \textbf{K}_2\)

—>> \(a \textbf{K} \times \textbf{K}_1 \leq \textbf{K} \times \textbf{K}_2 \leq b\textbf{K} \times \textbf{K}_1 \)

3. All norms defined on a finite dimensional space are \(\hspace{1.0cm}\).

functional

normed

—>> equivalent

linearised

4. If \(f : X \mapsto R\) be a linear functional defined on a linear space X, then f is a \(\hspace{1.0cm}\) function.

—>> convex

concave

divergent

convergent

5. If \(x^*\in R^n\) and if \(r>0\), then the \(\hspace{1.0cm}\) expressed as \(B(x^*, r) = \{y \in R^n: \textbf{K} y – x^* \textbf{K}<r\}\) centered at \(x^*\) of radius r is a convex set.

linear space

—>> ball

sphere

triangle

6. Any linear subspace M of \(R^n\) is a convex set since linear subspaces are \(\hspace{1.0cm}\) under addition and scalar multiplication.

normed

—>> closed

open

characterized

7. A nonempty subset C of the vector space X is convex if and only if C contains all \(\hspace{1.0cm}\) of all its elements.

—>> convex combinations

convex addition

convex multiplication

convex set

8. Let \(1 \leq p \leq + \infty\) . If for arbitrary \(x = \{x_k\}, y = \{y_k\} in \(l_p\) and \(\lambda \in K\), define vector addition and scalar multiplication componentwise then \(l_p\) is a \(\hspace{1.0cm}\) space.

nonlinear

—>> linear

real

complex

9. Let X be a linear space, and \(x, y \in X\). The [x, y] joining x and y is defined [x, y] = \(\{\lambda x + (1- \lambda)y: 0 \leq \lambda \leq 1\}\).

linear space

—>> line segment

subset

vector

10. The space of all bounded sequence of real(or complex) is called \(\hspace{1.0cm}\).

\(l_{1}\) -space

\(l_{0}\) -space

—>> \(l_{\infty}\) -space

\(l_{2}\) -space

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linear space

—>> line segment

subset

vector

nonlinear

—>> linear

real

complex

—>> convex combinations

convex addition

convex multiplication

convex set

normed

—>> closed

open

characterized

linear space

—>> ball

sphere

triangle

—>> convex

concave

divergent

convergent

functional

normed

—>> equivalent

linearised

universal

complement

union

—>> intersection

\(a \textbf{K} \times \textbf{K}_1 \geq 0\)

\(a \textbf{K} \times \textbf{K}_1 \geq \textbf{K} \times \textbf{K}_2\)

\(a \textbf{K} \times \textbf{K}_1 \leq \textbf{K} \times \textbf{K}_2\)

—>> \(a \textbf{K} \times \textbf{K}_1 \leq \textbf{K} \times \textbf{K}_2 \leq b\textbf{K} \times \textbf{K}_1 \)