QUIZ 4 (ANALYSIS II) Feb 14, 2020 (in tutorial) Name

QUIZ 4 (ANALYSIS II)

Feb 14, 2020 (in tutorial) Name:

In what follows,K∈ {R,C}. The symbolδij is the Kronecker symbol and feel free to ask the TA’s what it means if you don’t know. Finally,R+= [0,∞).

(1) Let (V,k k) be a finite dimensional normed linear space over K and W a vector subspace ofV. Show thatW is closed inV.

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(2) Leth, ibe an inner product on a finite dimensionalK-vector spaceV and letk k: V → R₊ be the usual norm, namely kvk = hv, vi¹². Let W be a subspace of V and w₁, . . . , w_k a basis of W such that hw_i, w_ji = δ_ij for i, j ∈ {1, . . . , k}. Let π: V →W be the map π(v) = Pk

i=1hv, wiiwi. Fix v ∈ V and define fv: W → R+ by the formula fv(w) = kw−vk. Show that

(a) fv is continuous onW;

(b) f_v attains its minimum at π(v) and if f_v(w) = f_v(π(v)) for some w∈W, thenw=π(v).

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