Telegram group: T.me/BBCNOUN [MTH412] Let \(x\) and \(v\) be vectors in \(R^{n}\), the line \(\L\) through \(x\) in the direction of \(v\) given by \(L={x+\alpha v: \alpha \epsilon R\) is a _________ convex set [MTH412] Any linear subspace \(M\) of \(R^{n}\) is a convex set since linear subspaces are________ under addition and scalar multiplication. closed [MTH412] Any linear subspace \(M\) of \(R^{n}\) is a convex set since linear subspaces are________ under addition and scalar multiplication. open [MTH412] Let \(p\leq 1\) be a fixed real number. Each element in the space \(l_{p}\) is a sequence, \(x = (x_{1}, x_{2}, . . ., x_{k}, . . .) of real numbers that converge then, \(sum_{k=1}^{\infty} |x_{k}|_{p}\) _________ n \(<\infty \) [MTH412] \(a k\times k_{2}\leq k\times k_{1}\) vector space [MTH412] Let X be a linear space and \(x,y \epsilon X \). The line segment [x, y] joining x and y is define by [x,y]= ___________ \({\lambda x +(1-\lambda)y:0 \leq \lambda \leq 1}\) [MTH412] Let X be a linear space and \(x,y \epsilon X \). The line segment [x, y] joining x and y is define by [x,y]= ___________ \({\lambda x +(1-\lambda)y:0 \leq \lambda \leq 1}\) [MTH412] The real line R becomes a normed linear space if\( k \times k\) is set to be__________ for every number \(x\epsilon R\). |x| [MTH412] Let \(k\cdot k_{1}\) and \(k\cdot k_{2}\) be two norms defined on a linear defined on a linear space \(X\cdot k \cdot }\) and \(k \cdot k_{2}\) are called equivalent if there exist constants a, b > 0 such that ______________ \(a k\times k_{1}\geq 0\) [MTH412] All norms defined on a finite dimensional space are ________________ normal