If X is the subspace of `∞ consisting of all sequences of zeros and ones, what is the induced metric on X ?

What is the definition of a bounded linear operator between Banach spaces, and what are the properties that characterize such operators?

3a

Find unitary oxthogonal to Mlowing diagonalizeny an or matr'y tor One 2 0 0 (as (b) 3 -2 - 2

Once you get away from vector spaces of finite dimension, it is no longer true that every linear map is automatically continuous. For example, let Coo {f E F(N, R) : f(n) = 0 for all sufficiently large n}, where F(N, R) is the vector space of all func- tions from N to R with the usual pointwise operations. Another way of expressing the condition for f to belong to Coo is that {n : f(n) # 0} is finite. Then define L: C00R by L(f) = Enf(n). Notice that the sum on the right hand side is in fact a finite sum. Now (a) Check that L is linear. (b) Show that {|L(f)| : ||F|| < 1} is not bounded above. Then deduce that L is not continuous, from the following theorem: For each continuous linear map L:V W the set {I|L(x)|lw: x|lv <1} is bounded above. 1 n=r (Hint: consider the functions e,, defined by e, (n) = and define a %3D 0 otherwise norm on e, by ||e,|| := sup{e,(n)|: n < 1} = le,(1)|. Show that for all r E N, |le,|| < 1. On the other hand {|L(e,)|:r€ N} is not bounded above.]

Prove that continuous image of pathwise connected space is pathwise connected

Give an example of 3 vector spaces that are not Rn. Explicitly state thedefinition of additon and the zero vector in each space. There is a solution on this site but it is hard to read and understand.

5. Determine whether the set of all ordered triples of real numbers with the op- erations defined below: (x, y, z) + (x',y',z') = (x+x',y+y',z+z') k(x, y, z) = (x, 1,z) is vector space or not.

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the ‚ a + 3b = c ·b +c+a = d contrary.

1D