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I have 2 questions -

  1. If $B = \{x(i) : i \in I \}$ be a basis of a subspace $S$ ( finite or infinite dimensional ) of a vector space $V$ and $C = \{ y(i) + S : i = 1 \text{ to } n \}$ is a basis of the finite dimensional space $V/S$ , then is it true that $B \cup \{ y(i) : i = 1 \text{ to } n \}$ is a basis to $V$?. If it is true what do I have to show to prove the statement ?

  2. Suppose $f : X \to R$ and $g : X \to R^n$ be 2 linear maps where kernel $g$ is a subset of kernel $f$ . I have shown that there exists a linear map $h : g(X) \to R$ such that $f = h(g)$. Can I extend h to R ^n without using Hann - Banach Theorems ?

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  1. Yes. Take $x\in X$, then $\hat{x}=\sum a_i\hat{y_i}$, which says $x-\sum a_i y_i\in S$. From here should be easy to show that your set is spanning. Linear independence is also immediate.

  2. Yes. $h(t):=f(x)$, where $x$ is such that $g(x)=t$. Linearity and the condition on the kernels ensure $h$ is well defined.

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