MTH412
Question: Let X be a linear space and \\(x, y\\in X\\). The ___________ [x, y] joining x and y is defined by [x, y] = \\( \\{\\lambda x + (1 – \\lambda)y : 0 \\leq \\lambda \\leq 1\\}\\)
Answer: line segment
Question: A linear space X, equipped with a norm k x k, is called a _________.
Answer: normed linear space
Question: Let D be a convex subset of a real vector space X and \\( f : D \\rightarrow R\\). Then \\( f (\\lambda x + (1 – \\lambda )y) < \\lambda f (x) + (1 – \\lambda)f(y)\\).
Answer: strictly convex
Question: Every norm induces a _______.
Answer: metric
Question: Let X be a linear space. A subset C of X is _________ if for every \\( x, y \\in C\\) and \\( \\lambda \\in [0, 1], \\lambda x + (1 – \\lambda)y \\in C\\).
Answer: convex
Question: Let A be a subset of a vector space X. The intersection of the (nonempty) family of all convex sets of X containing A is a convex set containing A and is obviously the smallest convex set of X containing A. This is called the ____ of A
Answer: convex hull
Question: Let D be a convex subset of a real vector space X and \\( f : D \\rightarrow R\\). Then \\( f (\\lambda x + (1 – \\lambda )y) > \\lambda f (x) + (1 – \\lambda)f(y)\\).
Answer: strictly concave
Question: Let \\(x_1, \\ldots x_n\\) be n points of the vector space X. Any elements of the form x = \\(\\sum_{i =1}^{n}\\lambda_i x_i\\) with \\(\\lambda_i \\geq 0 and \\(\\sum_{i =1}^{n}\\lambda_i = 1 \\) is called _____ of the elements \\(x_1, \\ldots x_n\\).
Answer: convex combination
Question: Let D be a convex subset of a real vector space X and \\( f : D \\rightarrow R\\). Then \\( f (\\lambda x + (1 – \\lambda )y) \\geq \\lambda f (x) + (1 – \\lambda)f(y)\\).
Answer: concave
Question: The formula \\(\\rho_{\\infty}(x, y) = Kx – yK_{\\infty} = sup_{k \\geq 1} \\mid x_k – y_k \\mid \\) induces a metric on the space \\(l _{\\infty}\\) of _________ sequences.
Answer: bounded
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