MTH412 Tma Solutions
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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